Ramamurthi fractions quadratic permutation results

by FEREGO Research

1. Introduction

Define the quadratic fraction form

                          P1(m) P2(n) - Q1(m) Q2(n)
  Zq({P1,Q1,R1,S1},{P2,Q2,R2,S2},m,n) = -------------------------
                         R1(m) R2(n) - S1(m) S2(n)
to be the fractional form for the given values of P, Q, R, and S. The dependence on the particular permutation {P, Q, R, S } is shown. In fact, this defined form

P, Q, R, and S are functions of a and b, and these are defined elsewhere. We use these same definitions.

Previous studies have found indices for m, and n such that the fraction results are equal. For example, we have

  Zq({P,Q,R,S},{P,Q,R,S},8,2) = Zq({P,Q,R,S},{P,Q,R,S},3,3)
from studies previously done.

The purpose of the present study is to find identity relationships for other permutations of the set {P, Q, R, S}. If P* and Q* are any permutation of {P, Q, R, S} then we find solutions m, n satisfying the equation

  Zq(P*,Q*,m,n) = Zq(P*,Q*,3,3) .
Note that the given permutations P* and Q* are the same on both sides of the equation. Also note that we are finding solutions that "match" the {3, 3} index.

Our results show that "nice" solutions exist for other permutations as well. These results are surprising and amazing. When we tabulate the results below, the bracketed pairs {m, n} are solutions to the above equation for the given permutations.

2. Results

These results were computed using a rather old (4 year old) Pentium computer, running at slightly over 1G hz. The computations on this page took several hours.

Searches here are limited, and all solutions may not be found. Work is being done in improving and checking the computations.

2.1. {P, Q, R, S} x {P, Q, R, S}

There are 15 solutions.

     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b
--------------------------------------------------------------------------
     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.2. {P, Q, R, S} x {P, Q, S, R}

There are 0 solutions.

2.3. {P, Q, R, S} x {P, R, Q, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.4. {P, Q, R, S} x {P, R, S, Q}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.5. {P, Q, R, S} x {P, S, Q, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3        2  4           5         6
  1980 a  + 5124 a  b - 3525 a  b  - 12870 a  b  + 555 a  b  + 6756 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.6. {P, Q, R, S} x {P, S, R, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.7. {P, Q, R, S} x {Q, P, R, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.8. {P, Q, R, S} x {Q, P, S, R}

There are 0 solutions.

2.9. {P, Q, R, S} x {Q, R, P, S}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.10. {P, Q, R, S} x {Q, R, S, P}

There are 6 solutions.

    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.11. {P, Q, R, S} x {Q, S, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.12. {P, Q, R, S} x {Q, S, R, P}

There are 6 solutions.

       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.13. {P, Q, R, S} x {R, P, Q, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 25062 a  b - 34745 a  b  + 16220 a  b  + 31165 a  b  + 1302 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.14. {P, Q, R, S} x {R, P, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.15. {P, Q, R, S} x {R, Q, P, S}

There are 6 solutions.

      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.16. {P, Q, R, S} x {R, Q, S, P}

There are 6 solutions.

         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.17. {P, Q, R, S} x {R, S, P, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.18. {P, Q, R, S} x {R, S, Q, P}

There are 6 solutions.

               2          2 3
         120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.19. {P, Q, R, S} x {S, P, Q, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.20. {P, Q, R, S} x {S, P, R, Q}

There are 6 solutions.

         6         5          4  2         3  3         2  4           5        6
 3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.21. {P, Q, R, S} x {S, Q, P, R}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.22. {P, Q, R, S} x {S, Q, R, P}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.23. {P, Q, R, S} x {S, R, P, Q}

There are 6 solutions.

             4      3          3      4
-343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.24. {P, Q, R, S} x {S, R, Q, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.25. {P, Q, S, R} x {P, Q, R, S}

There are 0 solutions.

2.26. {P, Q, S, R} x {P, Q, S, R}

There are 15 solutions.

     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b
---------------------------------------------------------------------------
      6       5           4  2         3  3        2  4          5        6
-120 a  - 74 a  b + 1615 a  b  + 2400 a  b  + 185 a  b  - 646 a b  - 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.27. {P, Q, S, R} x {P, R, Q, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.28. {P, Q, S, R} x {P, R, S, Q}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.29. {P, Q, S, R} x {P, S, Q, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.30. {P, Q, S, R} x {P, S, R, Q}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5           4  2         3  3        2  4           5        6
  3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.31. {P, Q, S, R} x {Q, P, R, S}

There are 0 solutions.

2.32. {P, Q, S, R} x {Q, P, S, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.33. {P, Q, S, R} x {Q, R, P, S}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
------------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.34. {P, Q, S, R} x {Q, R, S, P}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.35. {P, Q, S, R} x {Q, S, P, R}

There are 6 solutions.

      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.36. {P, Q, S, R} x {Q, S, R, P}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.37. {P, Q, S, R} x {R, P, Q, S}

There are 6 solutions.

      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.38. {P, Q, S, R} x {R, P, S, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.39. {P, Q, S, R} x {R, Q, P, S}

There are 6 solutions.

          6         5           4  2         3  3          2  4           5         6
-3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.40. {P, Q, S, R} x {R, Q, S, P}

There are 6 solutions.

        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.41. {P, Q, S, R} x {R, S, P, Q}

There are 6 solutions.

                2          2 3
         -120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.42. {P, Q, S, R} x {R, S, Q, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.43. {P, Q, S, R} x {S, P, Q, R}

There are 6 solutions.

          6         5          4  2         3  3         2  4           5        6
 -3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.44. {P, Q, S, R} x {S, P, R, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.45. {P, Q, S, R} x {S, Q, P, R}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.46. {P, Q, S, R} x {S, Q, R, P}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.47. {P, Q, S, R} x {S, R, P, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.48. {P, Q, S, R} x {S, R, Q, P}

There are 6 solutions.

            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
---------------------------------------
               2          2 3
         120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.49. {P, R, Q, S} x {P, Q, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.50. {P, R, Q, S} x {P, Q, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.51. {P, R, Q, S} x {P, R, Q, S}

There are 15 solutions.

  34
-(--)
  57

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.52. {P, R, Q, S} x {P, R, S, Q}

There are 0 solutions.

2.53. {P, R, Q, S} x {P, S, Q, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.54. {P, R, Q, S} x {P, S, R, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.55. {P, R, Q, S} x {Q, P, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.56. {P, R, Q, S} x {Q, P, S, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.57. {P, R, Q, S} x {Q, R, P, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.58. {P, R, Q, S} x {Q, R, S, P}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.59. {P, R, Q, S} x {Q, S, P, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.60. {P, R, Q, S} x {Q, S, R, P}

There are 6 solutions.

  2197
-(----)
  2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.61. {P, R, Q, S} x {R, P, Q, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.62. {P, R, Q, S} x {R, P, S, Q}

There are 0 solutions.

2.63. {P, R, Q, S} x {R, Q, P, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.64. {P, R, Q, S} x {R, Q, S, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.65. {P, R, Q, S} x {R, S, P, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.66. {P, R, Q, S} x {R, S, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.67. {P, R, Q, S} x {S, P, Q, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.68. {P, R, Q, S} x {S, P, R, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.69. {P, R, Q, S} x {S, Q, P, R}

There are 6 solutions.

2401
----
2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.70. {P, R, Q, S} x {S, Q, R, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.71. {P, R, Q, S} x {S, R, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.72. {P, R, Q, S} x {S, R, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.73. {P, R, S, Q} x {P, Q, R, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.74. {P, R, S, Q} x {P, Q, S, R}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.75. {P, R, S, Q} x {P, R, Q, S}

There are 0 solutions.

2.76. {P, R, S, Q} x {P, R, S, Q}

There are 15 solutions.

34
--
57

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.77. {P, R, S, Q} x {P, S, Q, R}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.78. {P, R, S, Q} x {P, S, R, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.79. {P, R, S, Q} x {Q, P, R, S}

There are 6 solutions.

      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.80. {P, R, S, Q} x {Q, P, S, R}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 25062 a  b - 34745 a  b  + 16220 a  b  + 31165 a  b  + 1302 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.81. {P, R, S, Q} x {Q, R, P, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.82. {P, R, S, Q} x {Q, R, S, P}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.83. {P, R, S, Q} x {Q, S, P, R}

There are 6 solutions.

2197
----
2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.84. {P, R, S, Q} x {Q, S, R, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.85. {P, R, S, Q} x {R, P, Q, S}

There are 0 solutions.

2.86. {P, R, S, Q} x {R, P, S, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.87. {P, R, S, Q} x {R, Q, P, S}

There are 6 solutions.

      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.88. {P, R, S, Q} x {R, Q, S, P}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.89. {P, R, S, Q} x {R, S, P, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 1302 a  b + 31165 a  b  + 16220 a  b  - 34745 a  b  - 25062 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.90. {P, R, S, Q} x {R, S, Q, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.91. {P, R, S, Q} x {S, P, Q, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.92. {P, R, S, Q} x {S, P, R, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.93. {P, R, S, Q} x {S, Q, P, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.94. {P, R, S, Q} x {S, Q, R, P}

There are 6 solutions.

  2401
-(----)
  2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.95. {P, R, S, Q} x {S, R, P, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.96. {P, R, S, Q} x {S, R, Q, P}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.97. {P, S, Q, R} x {P, Q, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3        2  4           5         6
  1980 a  + 5124 a  b - 3525 a  b  - 12870 a  b  + 555 a  b  + 6756 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.98. {P, S, Q, R} x {P, Q, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.99. {P, S, Q, R} x {P, R, Q, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.100. {P, S, Q, R} x {P, R, S, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.101. {P, S, Q, R} x {P, S, Q, R}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b
--------------------------------------------------------------------------------
      6         5           4  2          3  3         2  4           5        6
-840 a  - 3026 a  b + 5035 a  b  + 16800 a  b  + 7565 a  b  - 2014 a b  - 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.102. {P, S, Q, R} x {P, S, R, Q}

There are 0 solutions.

2.103. {P, S, Q, R} x {Q, P, R, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.104. {P, S, Q, R} x {Q, P, S, R}

There are 6 solutions.

         6         5          4  2         3  3         2  4           5        6
 3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.105. {P, S, Q, R} x {Q, R, P, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.106. {P, S, Q, R} x {Q, R, S, P}

There are 6 solutions.

             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.107. {P, S, Q, R} x {Q, S, P, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.108. {P, S, Q, R} x {Q, S, R, P}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.109. {P, S, Q, R} x {R, P, Q, S}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.110. {P, S, Q, R} x {R, P, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.111. {P, S, Q, R} x {R, Q, P, S}

There are 6 solutions.

                2          2 3
         -840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.112. {P, S, Q, R} x {R, Q, S, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.113. {P, S, Q, R} x {R, S, P, Q}

There are 6 solutions.

         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.114. {P, S, Q, R} x {R, S, Q, P}

There are 6 solutions.

        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.115. {P, S, Q, R} x {S, P, Q, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.116. {P, S, Q, R} x {S, P, R, Q}

There are 0 solutions.

2.117. {P, S, Q, R} x {S, Q, P, R}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
 -7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.118. {P, S, Q, R} x {S, Q, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.119. {P, S, Q, R} x {S, R, P, Q}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.120. {P, S, Q, R} x {S, R, Q, P}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.121. {P, S, R, Q} x {P, Q, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.122. {P, S, R, Q} x {P, Q, S, R}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5           4  2         3  3        2  4           5        6
  3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.123. {P, S, R, Q} x {P, R, Q, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.124. {P, S, R, Q} x {P, R, S, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.125. {P, S, R, Q} x {P, S, Q, R}

There are 0 solutions.

2.126. {P, S, R, Q} x {P, S, R, Q}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b
-------------------------------------------------------------------------------
     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.127. {P, S, R, Q} x {Q, P, R, S}

There are 6 solutions.

          6         5          4  2         3  3         2  4           5        6
 -3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.128. {P, S, R, Q} x {Q, P, S, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.129. {P, S, R, Q} x {Q, R, P, S}

There are 6 solutions.

              4      3          3      4
-2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
-----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.130. {P, S, R, Q} x {Q, R, S, P}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.131. {P, S, R, Q} x {Q, S, P, R}

There are 6 solutions.

        6       5            4  2          3  3          2  4            5         6
  3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 12768 a  b - 72905 a  b  - 71050 a  b  + 71155 a  b  + 70392 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.132. {P, S, R, Q} x {Q, S, R, P}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.133. {P, S, R, Q} x {R, P, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.134. {P, S, R, Q} x {R, P, S, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.135. {P, S, R, Q} x {R, Q, P, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.136. {P, S, R, Q} x {R, Q, S, P}

There are 6 solutions.

                2          2 3
          840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.137. {P, S, R, Q} x {R, S, P, Q}

There are 6 solutions.

      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.138. {P, S, R, Q} x {R, S, Q, P}

There are 6 solutions.

          6         5            4  2         3  3         2  4           5         6
-3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.139. {P, S, R, Q} x {S, P, Q, R}

There are 0 solutions.

2.140. {P, S, R, Q} x {S, P, R, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.141. {P, S, R, Q} x {S, Q, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.142. {P, S, R, Q} x {S, Q, R, P}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.143. {P, S, R, Q} x {S, R, P, Q}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.144. {P, S, R, Q} x {S, R, Q, P}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.145. {Q, P, R, S} x {P, Q, R, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.146. {Q, P, R, S} x {P, Q, S, R}

There are 0 solutions.

2.147. {Q, P, R, S} x {P, R, Q, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.148. {Q, P, R, S} x {P, R, S, Q}

There are 6 solutions.

      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.149. {Q, P, R, S} x {P, S, Q, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.150. {Q, P, R, S} x {P, S, R, Q}

There are 6 solutions.

          6         5          4  2         3  3         2  4           5        6
 -3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.151. {Q, P, R, S} x {Q, P, R, S}

There are 15 solutions.

     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b
---------------------------------------------------------------------------
      6       5           4  2         3  3        2  4          5        6
-120 a  - 74 a  b + 1615 a  b  + 2400 a  b  + 185 a  b  - 646 a b  - 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.152. {Q, P, R, S} x {Q, P, S, R}

There are 0 solutions.

2.153. {Q, P, R, S} x {Q, R, P, S}

There are 6 solutions.

        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.154. {Q, P, R, S} x {Q, R, S, P}

There are 6 solutions.

          6         5           4  2         3  3          2  4           5         6
-3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.155. {Q, P, R, S} x {Q, S, P, R}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.156. {Q, P, R, S} x {Q, S, R, P}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.157. {Q, P, R, S} x {R, P, Q, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.158. {Q, P, R, S} x {R, P, S, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.159. {Q, P, R, S} x {R, Q, P, S}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.160. {Q, P, R, S} x {R, Q, S, P}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
------------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.161. {Q, P, R, S} x {R, S, P, Q}

There are 6 solutions.

            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
---------------------------------------
               2          2 3
         120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.162. {Q, P, R, S} x {R, S, Q, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.163. {Q, P, R, S} x {S, P, Q, R}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5           4  2         3  3        2  4           5        6
  3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.164. {Q, P, R, S} x {S, P, R, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.165. {Q, P, R, S} x {S, Q, P, R}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.166. {Q, P, R, S} x {S, Q, R, P}

There are 6 solutions.

      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.167. {Q, P, R, S} x {S, R, P, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.168. {Q, P, R, S} x {S, R, Q, P}

There are 6 solutions.

                2          2 3
         -120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.169. {Q, P, S, R} x {P, Q, R, S}

There are 0 solutions.

2.170. {Q, P, S, R} x {P, Q, S, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.171. {Q, P, S, R} x {P, R, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.172. {Q, P, S, R} x {P, R, S, Q}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 25062 a  b - 34745 a  b  + 16220 a  b  + 31165 a  b  + 1302 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.173. {Q, P, S, R} x {P, S, Q, R}

There are 6 solutions.

         6         5          4  2         3  3         2  4           5        6
 3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.174. {Q, P, S, R} x {P, S, R, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.175. {Q, P, S, R} x {Q, P, R, S}

There are 0 solutions.

2.176. {Q, P, S, R} x {Q, P, S, R}

There are 15 solutions.

     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b
--------------------------------------------------------------------------
     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.177. {Q, P, S, R} x {Q, R, P, S}

There are 6 solutions.

         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.178. {Q, P, S, R} x {Q, R, S, P}

There are 6 solutions.

      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.179. {Q, P, S, R} x {Q, S, P, R}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.180. {Q, P, S, R} x {Q, S, R, P}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.181. {Q, P, S, R} x {R, P, Q, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.182. {Q, P, S, R} x {R, P, S, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.183. {Q, P, S, R} x {R, Q, P, S}

There are 6 solutions.

    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.184. {Q, P, S, R} x {R, Q, S, P}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.185. {Q, P, S, R} x {R, S, P, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.186. {Q, P, S, R} x {R, S, Q, P}

There are 6 solutions.

             4      3          3      4
-343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.187. {Q, P, S, R} x {S, P, Q, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.188. {Q, P, S, R} x {S, P, R, Q}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3        2  4           5         6
  1980 a  + 5124 a  b - 3525 a  b  - 12870 a  b  + 555 a  b  + 6756 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.189. {Q, P, S, R} x {S, Q, P, R}

There are 6 solutions.

       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.190. {Q, P, S, R} x {S, Q, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.191. {Q, P, S, R} x {S, R, P, Q}

There are 6 solutions.

               2          2 3
         120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.192. {Q, P, S, R} x {S, R, Q, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.193. {Q, R, P, S} x {P, Q, R, S}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.194. {Q, R, P, S} x {P, Q, S, R}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
------------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.195. {Q, R, P, S} x {P, R, Q, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.196. {Q, R, P, S} x {P, R, S, Q}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.197. {Q, R, P, S} x {P, S, Q, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.198. {Q, R, P, S} x {P, S, R, Q}

There are 6 solutions.

              4      3          3      4
-2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
-----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.199. {Q, R, P, S} x {Q, P, R, S}

There are 6 solutions.

        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.200. {Q, R, P, S} x {Q, P, S, R}

There are 6 solutions.

         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.201. {Q, R, P, S} x {Q, R, P, S}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b
--------------------------------------------------------------------------------
      6         5           4  2          3  3         2  4           5        6
-840 a  - 2014 a  b + 7565 a  b  + 16800 a  b  + 5035 a  b  - 3026 a b  - 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.202. {Q, R, P, S} x {Q, R, S, P}

There are 0 solutions.

2.203. {Q, R, P, S} x {Q, S, P, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.204. {Q, R, P, S} x {Q, S, R, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.205. {Q, R, P, S} x {R, P, Q, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.206. {Q, R, P, S} x {R, P, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.207. {Q, R, P, S} x {R, Q, P, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.208. {Q, R, P, S} x {R, Q, S, P}

There are 0 solutions.

2.209. {Q, R, P, S} x {R, S, P, Q}

There are 6 solutions.

         6         5           4  2         3  3        2  4           5        6
 3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.210. {Q, R, P, S} x {R, S, Q, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.211. {Q, R, P, S} x {S, P, Q, R}

There are 6 solutions.

                2          2 3
          840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.212. {Q, R, P, S} x {S, P, R, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.213. {Q, R, P, S} x {S, Q, P, R}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
 -7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.214. {Q, R, P, S} x {S, Q, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.215. {Q, R, P, S} x {S, R, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.216. {Q, R, P, S} x {S, R, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5          4  2          3  3         2  4           5         6
  1980 a  + 6756 a  b + 555 a  b  - 12870 a  b  - 3525 a  b  + 5124 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.217. {Q, R, S, P} x {P, Q, R, S}

There are 6 solutions.

    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.218. {Q, R, S, P} x {P, Q, S, R}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.219. {Q, R, S, P} x {P, R, Q, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.220. {Q, R, S, P} x {P, R, S, Q}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.221. {Q, R, S, P} x {P, S, Q, R}

There are 6 solutions.

             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.222. {Q, R, S, P} x {P, S, R, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.223. {Q, R, S, P} x {Q, P, R, S}

There are 6 solutions.

          6         5           4  2         3  3          2  4           5         6
-3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.224. {Q, R, S, P} x {Q, P, S, R}

There are 6 solutions.

      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.225. {Q, R, S, P} x {Q, R, P, S}

There are 0 solutions.

2.226. {Q, R, S, P} x {Q, R, S, P}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b
-------------------------------------------------------------------------------
     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.227. {Q, R, S, P} x {Q, S, P, R}

There are 6 solutions.

        6          5            4  2          3  3          2  4         5         6
  3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 70392 a  b + 71155 a  b  - 71050 a  b  - 72905 a  b  + 12768 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.228. {Q, R, S, P} x {Q, S, R, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.229. {Q, R, S, P} x {R, P, Q, S}

There are 6 solutions.

      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.230. {Q, R, S, P} x {R, P, S, Q}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.231. {Q, R, S, P} x {R, Q, P, S}

There are 0 solutions.

2.232. {Q, R, S, P} x {R, Q, S, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.233. {Q, R, S, P} x {R, S, P, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.234. {Q, R, S, P} x {R, S, Q, P}

There are 6 solutions.

          6         5           4  2         3  3        2  4           5        6
 -3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.235. {Q, R, S, P} x {S, P, Q, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.236. {Q, R, S, P} x {S, P, R, Q}

There are 6 solutions.

                2          2 3
         -840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.237. {Q, R, S, P} x {S, Q, P, R}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.238. {Q, R, S, P} x {S, Q, R, P}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.239. {Q, R, S, P} x {S, R, P, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5          4  2         3  3         2  4           5        6
  3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.240. {Q, R, S, P} x {S, R, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.241. {Q, S, P, R} x {P, Q, R, S}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.242. {Q, S, P, R} x {P, Q, S, R}

There are 6 solutions.

      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.243. {Q, S, P, R} x {P, R, Q, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.244. {Q, S, P, R} x {P, R, S, Q}

There are 6 solutions.

2197
----
2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.245. {Q, S, P, R} x {P, S, Q, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.246. {Q, S, P, R} x {P, S, R, Q}

There are 6 solutions.

        6       5            4  2          3  3          2  4            5         6
  3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 12768 a  b - 72905 a  b  - 71050 a  b  + 71155 a  b  + 70392 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.247. {Q, S, P, R} x {Q, P, R, S}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.248. {Q, S, P, R} x {Q, P, S, R}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.249. {Q, S, P, R} x {Q, R, P, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.250. {Q, S, P, R} x {Q, R, S, P}

There are 6 solutions.

        6          5            4  2          3  3          2  4         5         6
  3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 70392 a  b + 71155 a  b  - 71050 a  b  - 72905 a  b  + 12768 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.251. {Q, S, P, R} x {Q, S, P, R}

There are 15 solutions.

  57
-(--)
  34

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.252. {Q, S, P, R} x {Q, S, R, P}

There are 0 solutions.

2.253. {Q, S, P, R} x {R, P, Q, S}

There are 6 solutions.

  2401
-(----)
  2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.254. {Q, S, P, R} x {R, P, S, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.255. {Q, S, P, R} x {R, Q, P, S}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.256. {Q, S, P, R} x {R, Q, S, P}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.257. {Q, S, P, R} x {R, S, P, Q}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.258. {Q, S, P, R} x {R, S, Q, P}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.259. {Q, S, P, R} x {S, P, Q, R}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.260. {Q, S, P, R} x {S, P, R, Q}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.261. {Q, S, P, R} x {S, Q, P, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.262. {Q, S, P, R} x {S, Q, R, P}

There are 0 solutions.

2.263. {Q, S, P, R} x {S, R, P, Q}

There are 6 solutions.

      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.264. {Q, S, P, R} x {S, R, Q, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.265. {Q, S, R, P} x {P, Q, R, S}

There are 6 solutions.

       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.266. {Q, S, R, P} x {P, Q, S, R}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.267. {Q, S, R, P} x {P, R, Q, S}

There are 6 solutions.

  2197
-(----)
  2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.268. {Q, S, R, P} x {P, R, S, Q}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.269. {Q, S, R, P} x {P, S, Q, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.270. {Q, S, R, P} x {P, S, R, Q}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.271. {Q, S, R, P} x {Q, P, R, S}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.272. {Q, S, R, P} x {Q, P, S, R}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.273. {Q, S, R, P} x {Q, R, P, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.274. {Q, S, R, P} x {Q, R, S, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.275. {Q, S, R, P} x {Q, S, P, R}

There are 0 solutions.

2.276. {Q, S, R, P} x {Q, S, R, P}

There are 15 solutions.

57
--
34

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.277. {Q, S, R, P} x {R, P, Q, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.278. {Q, S, R, P} x {R, P, S, Q}

There are 6 solutions.

2401
----
2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.279. {Q, S, R, P} x {R, Q, P, S}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.280. {Q, S, R, P} x {R, Q, S, P}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
 -7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.281. {Q, S, R, P} x {R, S, P, Q}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.282. {Q, S, R, P} x {R, S, Q, P}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.283. {Q, S, R, P} x {S, P, Q, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.284. {Q, S, R, P} x {S, P, R, Q}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
 -7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.285. {Q, S, R, P} x {S, Q, P, R}

There are 0 solutions.

2.286. {Q, S, R, P} x {S, Q, R, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.287. {Q, S, R, P} x {S, R, P, Q}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.288. {Q, S, R, P} x {S, R, Q, P}

There are 6 solutions.

       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.289. {R, P, Q, S} x {P, Q, R, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 25062 a  b - 34745 a  b  + 16220 a  b  + 31165 a  b  + 1302 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.290. {R, P, Q, S} x {P, Q, S, R}

There are 6 solutions.

      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.291. {R, P, Q, S} x {P, R, Q, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.292. {R, P, Q, S} x {P, R, S, Q}

There are 0 solutions.

2.293. {R, P, Q, S} x {P, S, Q, R}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.294. {R, P, Q, S} x {P, S, R, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.295. {R, P, Q, S} x {Q, P, R, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.296. {R, P, Q, S} x {Q, P, S, R}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.297. {R, P, Q, S} x {Q, R, P, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.298. {R, P, Q, S} x {Q, R, S, P}

There are 6 solutions.

      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.299. {R, P, Q, S} x {Q, S, P, R}

There are 6 solutions.

  2401
-(----)
  2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.300. {R, P, Q, S} x {Q, S, R, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.301. {R, P, Q, S} x {R, P, Q, S}

There are 15 solutions.

34
--
57

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.302. {R, P, Q, S} x {R, P, S, Q}

There are 0 solutions.

2.303. {R, P, Q, S} x {R, Q, P, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.304. {R, P, Q, S} x {R, Q, S, P}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.305. {R, P, Q, S} x {R, S, P, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.306. {R, P, Q, S} x {R, S, Q, P}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.307. {R, P, Q, S} x {S, P, Q, R}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.308. {R, P, Q, S} x {S, P, R, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.309. {R, P, Q, S} x {S, Q, P, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.310. {R, P, Q, S} x {S, Q, R, P}

There are 6 solutions.

2197
----
2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.311. {R, P, Q, S} x {S, R, P, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.312. {R, P, Q, S} x {S, R, Q, P}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 1302 a  b + 31165 a  b  + 16220 a  b  - 34745 a  b  - 25062 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.313. {R, P, S, Q} x {P, Q, R, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.314. {R, P, S, Q} x {P, Q, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.315. {R, P, S, Q} x {P, R, Q, S}

There are 0 solutions.

2.316. {R, P, S, Q} x {P, R, S, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.317. {R, P, S, Q} x {P, S, Q, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.318. {R, P, S, Q} x {P, S, R, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.319. {R, P, S, Q} x {Q, P, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.320. {R, P, S, Q} x {Q, P, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.321. {R, P, S, Q} x {Q, R, P, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.322. {R, P, S, Q} x {Q, R, S, P}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.323. {R, P, S, Q} x {Q, S, P, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.324. {R, P, S, Q} x {Q, S, R, P}

There are 6 solutions.

2401
----
2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.325. {R, P, S, Q} x {R, P, Q, S}

There are 0 solutions.

2.326. {R, P, S, Q} x {R, P, S, Q}

There are 15 solutions.

  34
-(--)
  57

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.327. {R, P, S, Q} x {R, Q, P, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.328. {R, P, S, Q} x {R, Q, S, P}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.329. {R, P, S, Q} x {R, S, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.330. {R, P, S, Q} x {R, S, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.331. {R, P, S, Q} x {S, P, Q, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.332. {R, P, S, Q} x {S, P, R, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.333. {R, P, S, Q} x {S, Q, P, R}

There are 6 solutions.

  2197
-(----)
  2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.334. {R, P, S, Q} x {S, Q, R, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.335. {R, P, S, Q} x {S, R, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.336. {R, P, S, Q} x {S, R, Q, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.337. {R, Q, P, S} x {P, Q, R, S}

There are 6 solutions.

      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.338. {R, Q, P, S} x {P, Q, S, R}

There are 6 solutions.

          6         5           4  2         3  3          2  4           5         6
-3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.339. {R, Q, P, S} x {P, R, Q, S}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.340. {R, Q, P, S} x {P, R, S, Q}

There are 6 solutions.

      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b
--------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.341. {R, Q, P, S} x {P, S, Q, R}

There are 6 solutions.

                2          2 3
         -840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.342. {R, Q, P, S} x {P, S, R, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.343. {R, Q, P, S} x {Q, P, R, S}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.344. {R, Q, P, S} x {Q, P, S, R}

There are 6 solutions.

    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.345. {R, Q, P, S} x {Q, R, P, S}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.346. {R, Q, P, S} x {Q, R, S, P}

There are 0 solutions.

2.347. {R, Q, P, S} x {Q, S, P, R}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.348. {R, Q, P, S} x {Q, S, R, P}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.349. {R, Q, P, S} x {R, P, Q, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.350. {R, Q, P, S} x {R, P, S, Q}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.351. {R, Q, P, S} x {R, Q, P, S}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b
-------------------------------------------------------------------------------
     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.352. {R, Q, P, S} x {R, Q, S, P}

There are 0 solutions.

2.353. {R, Q, P, S} x {R, S, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.354. {R, Q, P, S} x {R, S, Q, P}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5          4  2         3  3         2  4           5        6
  3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.355. {R, Q, P, S} x {S, P, Q, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.356. {R, Q, P, S} x {S, P, R, Q}

There are 6 solutions.

             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.357. {R, Q, P, S} x {S, Q, P, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.358. {R, Q, P, S} x {S, Q, R, P}

There are 6 solutions.

        6          5            4  2          3  3          2  4         5         6
  3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 70392 a  b + 71155 a  b  - 71050 a  b  - 72905 a  b  + 12768 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.359. {R, Q, P, S} x {S, R, P, Q}

There are 6 solutions.

          6         5           4  2         3  3        2  4           5        6
 -3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.360. {R, Q, P, S} x {S, R, Q, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.361. {R, Q, S, P} x {P, Q, R, S}

There are 6 solutions.

         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.362. {R, Q, S, P} x {P, Q, S, R}

There are 6 solutions.

        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.363. {R, Q, S, P} x {P, R, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.364. {R, Q, S, P} x {P, R, S, Q}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.365. {R, Q, S, P} x {P, S, Q, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.366. {R, Q, S, P} x {P, S, R, Q}

There are 6 solutions.

                2          2 3
          840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.367. {R, Q, S, P} x {Q, P, R, S}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
------------------------------------
    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.368. {R, Q, S, P} x {Q, P, S, R}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.369. {R, Q, S, P} x {Q, R, P, S}

There are 0 solutions.

2.370. {R, Q, S, P} x {Q, R, S, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.371. {R, Q, S, P} x {Q, S, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.372. {R, Q, S, P} x {Q, S, R, P}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
 -7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.373. {R, Q, S, P} x {R, P, Q, S}

There are 6 solutions.

          6         5            4  2          3  3          2  4            5         6
-5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.374. {R, Q, S, P} x {R, P, S, Q}

There are 6 solutions.

         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.375. {R, Q, S, P} x {R, Q, P, S}

There are 0 solutions.

2.376. {R, Q, S, P} x {R, Q, S, P}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b
--------------------------------------------------------------------------------
      6         5           4  2          3  3         2  4           5        6
-840 a  - 2014 a  b + 7565 a  b  + 16800 a  b  + 5035 a  b  - 3026 a b  - 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.377. {R, Q, S, P} x {R, S, P, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5          4  2          3  3         2  4           5         6
  1980 a  + 6756 a  b + 555 a  b  - 12870 a  b  - 3525 a  b  + 5124 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.378. {R, Q, S, P} x {R, S, Q, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.379. {R, Q, S, P} x {S, P, Q, R}

There are 6 solutions.

              4      3          3      4
-2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
-----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.380. {R, Q, S, P} x {S, P, R, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.381. {R, Q, S, P} x {S, Q, P, R}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.382. {R, Q, S, P} x {S, Q, R, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.383. {R, Q, S, P} x {S, R, P, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.384. {R, Q, S, P} x {S, R, Q, P}

There are 6 solutions.

         6         5           4  2         3  3        2  4           5        6
 3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.385. {R, S, P, Q} x {P, Q, R, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.386. {R, S, P, Q} x {P, Q, S, R}

There are 6 solutions.

                2          2 3
         -120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.387. {R, S, P, Q} x {P, R, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.388. {R, S, P, Q} x {P, R, S, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 1302 a  b + 31165 a  b  + 16220 a  b  - 34745 a  b  - 25062 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.389. {R, S, P, Q} x {P, S, Q, R}

There are 6 solutions.

         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.390. {R, S, P, Q} x {P, S, R, Q}

There are 6 solutions.

      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.391. {R, S, P, Q} x {Q, P, R, S}

There are 6 solutions.

            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
---------------------------------------
               2          2 3
         120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.392. {R, S, P, Q} x {Q, P, S, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.393. {R, S, P, Q} x {Q, R, P, S}

There are 6 solutions.

         6         5           4  2         3  3        2  4           5        6
 3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.394. {R, S, P, Q} x {Q, R, S, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.395. {R, S, P, Q} x {Q, S, P, R}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.396. {R, S, P, Q} x {Q, S, R, P}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.397. {R, S, P, Q} x {R, P, Q, S}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.398. {R, S, P, Q} x {R, P, S, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.399. {R, S, P, Q} x {R, Q, P, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.400. {R, S, P, Q} x {R, Q, S, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5          4  2          3  3         2  4           5         6
  1980 a  + 6756 a  b + 555 a  b  - 12870 a  b  - 3525 a  b  + 5124 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.401. {R, S, P, Q} x {R, S, P, Q}

There are 15 solutions.

     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b
--------------------------------------------------------------------------
     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.402. {R, S, P, Q} x {R, S, Q, P}

There are 0 solutions.

2.403. {R, S, P, Q} x {S, P, Q, R}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.404. {R, S, P, Q} x {S, P, R, Q}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.405. {R, S, P, Q} x {S, Q, P, R}

There are 6 solutions.

       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.406. {R, S, P, Q} x {S, Q, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.407. {R, S, P, Q} x {S, R, P, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.408. {R, S, P, Q} x {S, R, Q, P}

There are 0 solutions.

2.409. {R, S, Q, P} x {P, Q, R, S}

There are 6 solutions.

               2          2 3
         120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.410. {R, S, Q, P} x {P, Q, S, R}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.411. {R, S, Q, P} x {P, R, Q, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.412. {R, S, Q, P} x {P, R, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.413. {R, S, Q, P} x {P, S, Q, R}

There are 6 solutions.

        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.414. {R, S, Q, P} x {P, S, R, Q}

There are 6 solutions.

          6         5            4  2         3  3         2  4           5         6
-3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.415. {R, S, Q, P} x {Q, P, R, S}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.416. {R, S, Q, P} x {Q, P, S, R}

There are 6 solutions.

             4      3          3      4
-343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.417. {R, S, Q, P} x {Q, R, P, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.418. {R, S, Q, P} x {Q, R, S, P}

There are 6 solutions.

          6         5           4  2         3  3        2  4           5        6
 -3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.419. {R, S, Q, P} x {Q, S, P, R}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.420. {R, S, Q, P} x {Q, S, R, P}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.421. {R, S, Q, P} x {R, P, Q, S}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.422. {R, S, Q, P} x {R, P, S, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.423. {R, S, Q, P} x {R, Q, P, S}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5          4  2         3  3         2  4           5        6
  3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.424. {R, S, Q, P} x {R, Q, S, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.425. {R, S, Q, P} x {R, S, P, Q}

There are 0 solutions.

2.426. {R, S, Q, P} x {R, S, Q, P}

There are 15 solutions.

     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b
---------------------------------------------------------------------------
      6        5          4  2         3  3         2  4         5        6
-120 a  - 646 a  b + 185 a  b  + 2400 a  b  + 1615 a  b  - 74 a b  - 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.427. {R, S, Q, P} x {S, P, Q, R}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.428. {R, S, Q, P} x {S, P, R, Q}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.429. {R, S, Q, P} x {S, Q, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.430. {R, S, Q, P} x {S, Q, R, P}

There are 6 solutions.

      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.431. {R, S, Q, P} x {S, R, P, Q}

There are 0 solutions.

2.432. {R, S, Q, P} x {S, R, Q, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.433. {S, P, Q, R} x {P, Q, R, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.434. {S, P, Q, R} x {P, Q, S, R}

There are 6 solutions.

          6         5          4  2         3  3         2  4           5        6
 -3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.435. {S, P, Q, R} x {P, R, Q, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.436. {S, P, Q, R} x {P, R, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.437. {S, P, Q, R} x {P, S, Q, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.438. {S, P, Q, R} x {P, S, R, Q}

There are 0 solutions.

2.439. {S, P, Q, R} x {Q, P, R, S}

There are 6 solutions.

       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5           4  2         3  3        2  4           5        6
  3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.440. {S, P, Q, R} x {Q, P, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.441. {S, P, Q, R} x {Q, R, P, S}

There are 6 solutions.

                2          2 3
          840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.442. {S, P, Q, R} x {Q, R, S, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.443. {S, P, Q, R} x {Q, S, P, R}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.444. {S, P, Q, R} x {Q, S, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.445. {S, P, Q, R} x {R, P, Q, S}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.446. {S, P, Q, R} x {R, P, S, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.447. {S, P, Q, R} x {R, Q, P, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.448. {S, P, Q, R} x {R, Q, S, P}

There are 6 solutions.

              4      3          3      4
-2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
-----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.449. {S, P, Q, R} x {R, S, P, Q}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.450. {S, P, Q, R} x {R, S, Q, P}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.451. {S, P, Q, R} x {S, P, Q, R}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b
-------------------------------------------------------------------------------
     6         5           4  2          3  3         2  4           5        6
840 a  + 3026 a  b - 5035 a  b  - 16800 a  b  - 7565 a  b  + 2014 a b  + 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.452. {S, P, Q, R} x {S, P, R, Q}

There are 0 solutions.

2.453. {S, P, Q, R} x {S, Q, P, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.454. {S, P, Q, R} x {S, Q, R, P}

There are 6 solutions.

        6       5            4  2          3  3          2  4            5         6
  3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 12768 a  b - 72905 a  b  - 71050 a  b  + 71155 a  b  + 70392 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.455. {S, P, Q, R} x {S, R, P, Q}

There are 6 solutions.

          6         5            4  2         3  3         2  4           5         6
-3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.456. {S, P, Q, R} x {S, R, Q, P}

There are 6 solutions.

      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.457. {S, P, R, Q} x {P, Q, R, S}

There are 6 solutions.

         6         5          4  2         3  3         2  4           5        6
 3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.458. {S, P, R, Q} x {P, Q, S, R}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.459. {S, P, R, Q} x {P, R, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.460. {S, P, R, Q} x {P, R, S, Q}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )
-----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.461. {S, P, R, Q} x {P, S, Q, R}

There are 0 solutions.

2.462. {S, P, R, Q} x {P, S, R, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.463. {S, P, R, Q} x {Q, P, R, S}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.464. {S, P, R, Q} x {Q, P, S, R}

There are 6 solutions.

      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3        2  4           5         6
  1980 a  + 5124 a  b - 3525 a  b  - 12870 a  b  + 555 a  b  + 6756 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.465. {S, P, R, Q} x {Q, R, P, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.466. {S, P, R, Q} x {Q, R, S, P}

There are 6 solutions.

                2          2 3
         -840 (a  + a b + b )
----------------------------------------
             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.467. {S, P, R, Q} x {Q, S, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.468. {S, P, R, Q} x {Q, S, R, P}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
 -7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.469. {S, P, R, Q} x {R, P, Q, S}

There are 6 solutions.

          6          5            4  2          3  3          2  4           5         6
-5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
-----------------------------------------------------------------------------------------
          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.470. {S, P, R, Q} x {R, P, S, Q}

There are 6 solutions.

         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.471. {S, P, R, Q} x {R, Q, P, S}

There are 6 solutions.

             4      3          3      4
2197 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          840 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.472. {S, P, R, Q} x {R, Q, S, P}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.473. {S, P, R, Q} x {R, S, P, Q}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.474. {S, P, R, Q} x {R, S, Q, P}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.475. {S, P, R, Q} x {S, P, Q, R}

There are 0 solutions.

2.476. {S, P, R, Q} x {S, P, R, Q}

There are 15 solutions.

     6         5           4  2          3  3         2  4           5        6
840 a  + 2014 a  b - 7565 a  b  - 16800 a  b  - 5035 a  b  + 3026 a b  + 840 b
--------------------------------------------------------------------------------
      6         5           4  2          3  3         2  4           5        6
-840 a  - 3026 a  b + 5035 a  b  + 16800 a  b  + 7565 a  b  - 2014 a b  - 840 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.477. {S, P, R, Q} x {S, Q, P, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.478. {S, P, R, Q} x {S, Q, R, P}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.479. {S, P, R, Q} x {S, R, P, Q}

There are 6 solutions.

        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.480. {S, P, R, Q} x {S, R, Q, P}

There are 6 solutions.

         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.481. {S, Q, P, R} x {P, Q, R, S}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.482. {S, Q, P, R} x {P, Q, S, R}

There are 6 solutions.

         6         5           4  2          3  3          2  4           5        6
-5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
-------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.483. {S, Q, P, R} x {P, R, Q, S}

There are 6 solutions.

2401
----
2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.484. {S, Q, P, R} x {P, R, S, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.485. {S, Q, P, R} x {P, S, Q, R}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
 -7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.486. {S, Q, P, R} x {P, S, R, Q}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 359 a  b + 631 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.487. {S, Q, P, R} x {Q, P, R, S}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.488. {S, Q, P, R} x {Q, P, S, R}

There are 6 solutions.

       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
-------------------------------------------------------------------------------------
       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.489. {S, Q, P, R} x {Q, R, P, S}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
 -7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.490. {S, Q, P, R} x {Q, R, S, P}

There are 6 solutions.

     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.491. {S, Q, P, R} x {Q, S, P, R}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.492. {S, Q, P, R} x {Q, S, R, P}

There are 0 solutions.

2.493. {S, Q, P, R} x {R, P, Q, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.494. {S, Q, P, R} x {R, P, S, Q}

There are 6 solutions.

  2197
-(----)
  2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.495. {S, Q, P, R} x {R, Q, P, S}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.496. {S, Q, P, R} x {R, Q, S, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
 -(3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
7 (1980 a  + 10056 a  b + 10165 a  b  - 10150 a  b  - 10415 a  b  + 1824 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.497. {S, Q, P, R} x {R, S, P, Q}

There are 6 solutions.

       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.498. {S, Q, P, R} x {R, S, Q, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.499. {S, Q, P, R} x {S, P, Q, R}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.500. {S, Q, P, R} x {S, P, R, Q}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
 -(3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
7 (1980 a  + 1824 a  b - 10415 a  b  - 10150 a  b  + 10165 a  b  + 10056 a b  + 1980 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.501. {S, Q, P, R} x {S, Q, P, R}

There are 15 solutions.

57
--
34

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.502. {S, Q, P, R} x {S, Q, R, P}

There are 0 solutions.

2.503. {S, Q, P, R} x {S, R, P, Q}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.504. {S, Q, P, R} x {S, R, Q, P}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.505. {S, Q, R, P} x {P, Q, R, S}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.506. {S, Q, R, P} x {P, Q, S, R}

There are 6 solutions.

        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.507. {S, Q, R, P} x {P, R, Q, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.508. {S, Q, R, P} x {P, R, S, Q}

There are 6 solutions.

  2401
-(----)
  2197

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.509. {S, Q, R, P} x {P, S, Q, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.510. {S, Q, R, P} x {P, S, R, Q}

There are 6 solutions.

          6         5           4  2          3  3          2  4           5        6
  7 (540 a  + 5736 a  b + 1525 a  b  - 20230 a  b  - 19055 a  b  - 2496 a b  + 540 b )
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.511. {S, Q, R, P} x {Q, P, R, S}

There are 6 solutions.

      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b
------------------------------------------------------------------------------------
      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.512. {S, Q, R, P} x {Q, P, S, R}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.513. {S, Q, R, P} x {Q, R, P, S}

There are 6 solutions.

     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.514. {S, Q, R, P} x {Q, R, S, P}

There are 6 solutions.

          6         5            4  2          3  3         2  4           5        6
  7 (540 a  - 2496 a  b - 19055 a  b  - 20230 a  b  + 1525 a  b  + 5736 a b  + 540 b )
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.515. {S, Q, R, P} x {Q, S, P, R}

There are 0 solutions.

2.516. {S, Q, R, P} x {Q, S, R, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.517. {S, Q, R, P} x {R, P, Q, S}

There are 6 solutions.

2197
----
2401

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.518. {S, Q, R, P} x {R, P, S, Q}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.519. {S, Q, R, P} x {R, Q, P, S}

There are 6 solutions.

        6          5            4  2          3  3          2  4         5         6
  3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 70392 a  b + 71155 a  b  - 71050 a  b  - 72905 a  b  + 12768 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.520. {S, Q, R, P} x {R, Q, S, P}

There are 6 solutions.

         6          5            4  2          3  3          2  4         5         6
   3780 a  + 22656 a  b + 83975 a  b  + 92470 a  b  + 27395 a  b  + 24 a b  + 3780 b
----------------------------------------------------------------------------------------
         6          5            4  2          3  3          2  4           5         6
5 (2772 a  + 15384 a  b + 13687 a  b  - 21826 a  b  - 21653 a  b  + 1248 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.521. {S, Q, R, P} x {R, S, P, Q}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.522. {S, Q, R, P} x {R, S, Q, P}

There are 6 solutions.

      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.523. {S, Q, R, P} x {S, P, Q, R}

There are 6 solutions.

        6       5            4  2          3  3          2  4            5         6
  3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
---------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4            5          6
13860 a  + 12768 a  b - 72905 a  b  - 71050 a  b  + 71155 a  b  + 70392 a b  + 13860 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.524. {S, Q, R, P} x {S, P, R, Q}

There are 6 solutions.

         6       5            4  2          3  3          2  4            5         6
   3780 a  + 24 a  b + 27395 a  b  + 92470 a  b  + 83975 a  b  + 22656 a b  + 3780 b
----------------------------------------------------------------------------------------
         6         5            4  2          3  3          2  4            5         6
5 (2772 a  + 1248 a  b - 21653 a  b  - 21826 a  b  + 13687 a  b  + 15384 a b  + 2772 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.525. {S, Q, R, P} x {S, Q, P, R}

There are 0 solutions.

2.526. {S, Q, R, P} x {S, Q, R, P}

There are 15 solutions.

  57
-(--)
  34

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.527. {S, Q, R, P} x {S, R, P, Q}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.528. {S, Q, R, P} x {S, R, Q, P}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.529. {S, R, P, Q} x {P, Q, R, S}

There are 6 solutions.

             4      3          3      4
-343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
----------------------------------------
                2          2 3
          120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.530. {S, R, P, Q} x {P, Q, S, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.531. {S, R, P, Q} x {P, R, Q, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
      6         5           4  2          3  3          2  4            5         6
1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.532. {S, R, P, Q} x {P, R, S, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.533. {S, R, P, Q} x {P, S, Q, R}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.534. {S, R, P, Q} x {P, S, R, Q}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
-11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.535. {S, R, P, Q} x {Q, P, R, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.536. {S, R, P, Q} x {Q, P, S, R}

There are 6 solutions.

               2          2 3
         120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.537. {S, R, P, Q} x {Q, R, P, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3900 a  - 14692 a  b + 18125 a  b  + 70710 a  b  + 33085 a  b  - 8708 a b  - 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.538. {S, R, P, Q} x {Q, R, S, P}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3900 a  - 8708 a  b + 33085 a  b  + 70710 a  b  + 18125 a  b  - 14692 a b  - 3900 b
-------------------------------------------------------------------------------------
          6         5          4  2         3  3         2  4           5        6
  3 (660 a  + 2252 a  b + 185 a  b  - 4290 a  b  - 1175 a  b  + 1708 a b  + 660 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.539. {S, R, P, Q} x {Q, S, P, R}

There are 6 solutions.

      6          5            4  2          3  3         2  4           5         6
1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.540. {S, R, P, Q} x {Q, S, R, P}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.541. {S, R, P, Q} x {R, P, Q, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
-90 a  + 701 a  b + 971 a b  + 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.542. {S, R, P, Q} x {R, P, S, Q}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 1302 a  b - 31165 a  b  - 16220 a  b  + 34745 a  b  + 25062 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.543. {S, R, P, Q} x {R, Q, P, S}

There are 6 solutions.

          6         5           4  2         3  3        2  4           5        6
 -3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.544. {S, R, P, Q} x {R, Q, S, P}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
--------------------------------------
      3        2            2        3
-330 a  - 631 a  b + 359 a b  + 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.545. {S, R, P, Q} x {R, S, P, Q}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.546. {S, R, P, Q} x {R, S, Q, P}

There are 0 solutions.

2.547. {S, R, P, Q} x {S, P, Q, R}

There are 6 solutions.

          6         5            4  2         3  3         2  4           5         6
-3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
--------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.548. {S, R, P, Q} x {S, P, R, Q}

There are 6 solutions.

        6         5            4  2         3  3         2  4           5         6
  1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
-(---------------------------------------------------------------------------------)
        6         5           4  2         3  3          2  4           5         6
  1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.549. {S, R, P, Q} x {S, Q, P, R}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.550. {S, R, P, Q} x {S, Q, R, P}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.551. {S, R, P, Q} x {S, R, P, Q}

There are 15 solutions.

     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b
---------------------------------------------------------------------------
      6        5          4  2         3  3         2  4         5        6
-120 a  - 646 a  b + 185 a  b  + 2400 a  b  + 1615 a  b  - 74 a b  - 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

2.552. {S, R, P, Q} x {S, R, Q, P}

There are 0 solutions.

2.553. {S, R, Q, P} x {P, Q, R, S}

There are 10 solutions.

-1

{{3, 3}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.554. {S, R, Q, P} x {P, Q, S, R}

There are 6 solutions.

            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )
---------------------------------------
               2          2 3
         120 (a  + a b + b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.555. {S, R, Q, P} x {P, R, Q, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3960 a  - 7014 a  b - 54965 a  b  - 35260 a  b  + 39505 a  b  + 30774 a b  + 3960 b
------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.556. {S, R, Q, P} x {P, R, S, Q}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 7014 a  b + 54965 a  b  + 35260 a  b  - 39505 a  b  - 30774 a b  - 3960 b
-------------------------------------------------------------------------------------
       6         5           4  2          3  3          2  4            5         6
 1080 a  - 9942 a  b - 4505 a  b  + 51500 a  b  + 61405 a  b  + 16422 a b  + 1080 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.557. {S, R, Q, P} x {P, S, Q, R}

There are 6 solutions.

          6         5           4  2         3  3         2  4           5        6
 11 (180 a  + 2212 a  b + 4045 a  b  - 1170 a  b  - 4315 a  b  - 1132 a b  + 180 b )
-------------------------------------------------------------------------------------
         6         5           4  2         3  3          2  4           5         6
3 (1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.558. {S, R, Q, P} x {P, S, R, Q}

There are 6 solutions.

    3        2            2       3
90 a  + 971 a  b + 701 a b  - 90 b
-----------------------------------
    3        2            2       3
90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.559. {S, R, Q, P} x {Q, P, R, S}

There are 6 solutions.

                2          2 3
         -120 (a  + a b + b )
---------------------------------------
            4      3          3      4
343 a b (2 a  + 5 a  b - 5 a b  - 2 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.560. {S, R, Q, P} x {Q, P, S, R}

There are 12 solutions.

1

{{2, 2}, {3, 3}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.561. {S, R, Q, P} x {Q, R, P, S}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
        6         5          4  2          3  3         2  4           5         6
  1980 a  + 6756 a  b + 555 a  b  - 12870 a  b  - 3525 a  b  + 5124 a b  + 1980 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.562. {S, R, Q, P} x {Q, R, S, P}

There are 6 solutions.

      6         5            4  2          3  3          2  4            5         6
3900 a  + 8708 a  b - 33085 a  b  - 70710 a  b  - 18125 a  b  + 14692 a b  + 3900 b
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.563. {S, R, Q, P} x {Q, S, P, R}

There are 6 solutions.

     3        2            2       3
 90 a  + 971 a  b + 701 a b  - 90 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.564. {S, R, Q, P} x {Q, S, R, P}

There are 6 solutions.

       6          5            4  2          3  3         2  4           5         6
 1080 a  + 16422 a  b + 61405 a  b  + 51500 a  b  - 4505 a  b  - 9942 a b  + 1080 b
-------------------------------------------------------------------------------------
       6          5            4  2          3  3          2  4           5         6
-3960 a  - 30774 a  b - 39505 a  b  + 35260 a  b  + 54965 a  b  + 7014 a b  - 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.565. {S, R, Q, P} x {R, P, Q, S}

There are 6 solutions.

       6         5            4  2          3  3          2  4            5         6
-3960 a  + 1302 a  b + 31165 a  b  + 16220 a  b  - 34745 a  b  - 25062 a b  - 3960 b
-------------------------------------------------------------------------------------
        6         5           4  2          3  3          2  4           5        6
5 (216 a  + 5034 a  b + 4951 a  b  - 13108 a  b  - 16979 a  b  - 3738 a b  + 216 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.566. {S, R, Q, P} x {R, P, S, Q}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2       3
 90 a  - 701 a  b - 971 a b  - 90 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.567. {S, R, Q, P} x {R, Q, P, S}

There are 6 solutions.

     3        2            2        3
330 a  + 359 a  b - 631 a b  - 330 b
-------------------------------------
     3        2            2        3
330 a  + 631 a  b - 359 a b  - 330 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.568. {S, R, Q, P} x {R, Q, S, P}

There are 6 solutions.

         6         5           4  2         3  3        2  4           5        6
 3 (660 a  + 1708 a  b - 1175 a  b  - 4290 a  b  + 185 a  b  + 2252 a b  + 660 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3900 a  + 14692 a  b - 18125 a  b  - 70710 a  b  - 33085 a  b  + 8708 a b  + 3900 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.569. {S, R, Q, P} x {R, S, P, Q}

There are 0 solutions.

2.570. {S, R, Q, P} x {R, S, Q, P}

There are 15 solutions.

0

{{2, 0}, {2, 2}, {3, 3}, {4, 0}, {4, 2}, {4, 4}, {5, 3}, {5, 5}, {6, 6}, {7, 3}, {7, 5}, {7, 7}, {8, 8}, {9, 9}, {10, 10}}

2.571. {S, R, Q, P} x {S, P, Q, R}

There are 6 solutions.

      6         5            4  2         3  3         2  4           5         6
1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b
---------------------------------------------------------------------------------
      6         5           4  2         3  3          2  4           5         6
1300 a  + 2228 a  b + 2405 a  b  + 6670 a  b  + 10765 a  b  + 5572 a b  + 1300 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.572. {S, R, Q, P} x {S, P, R, Q}

There are 6 solutions.

         6         5            4  2         3  3         2  4           5         6
3 (1300 a  + 5572 a  b + 10765 a  b  + 6670 a  b  + 2405 a  b  + 2228 a b  + 1300 b )
-------------------------------------------------------------------------------------
          6         5           4  2         3  3         2  4           5        6
 11 (180 a  - 1132 a  b - 4315 a  b  - 1170 a  b  + 4045 a  b  + 2212 a b  + 180 b )

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.573. {S, R, Q, P} x {S, Q, P, R}

There are 6 solutions.

         6         5            4  2          3  3         2  4           5        6
-5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
-------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 25062 a  b + 34745 a  b  - 16220 a  b  - 31165 a  b  - 1302 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.574. {S, R, Q, P} x {S, Q, R, P}

There are 6 solutions.

        6         5            4  2          3  3         2  4           5        6
5 (216 a  - 3738 a  b - 16979 a  b  - 13108 a  b  + 4951 a  b  + 5034 a b  + 216 b )
------------------------------------------------------------------------------------
      6          5            4  2          3  3          2  4           5         6
3960 a  + 30774 a  b + 39505 a  b  - 35260 a  b  - 54965 a  b  - 7014 a b  + 3960 b

{{3, 3}, {5, 3}, {5, 5}, {7, 3}, {7, 5}, {7, 7}}

2.575. {S, R, Q, P} x {S, R, P, Q}

There are 0 solutions.

2.576. {S, R, Q, P} x {S, R, Q, P}

There are 15 solutions.

     6       5           4  2         3  3        2  4          5        6
120 a  + 74 a  b - 1615 a  b  - 2400 a  b  - 185 a  b  + 646 a b  + 120 b
--------------------------------------------------------------------------
     6        5          4  2         3  3         2  4         5        6
120 a  + 646 a  b - 185 a  b  - 2400 a  b  - 1615 a  b  + 74 a b  + 120 b

{{3, 3}, {5, 3}, {5, 5}, {6, 0}, {6, 2}, {6, 4}, {7, 3}, {7, 5}, {7, 7}, {8, 0}, {8, 2}, {8, 4}, {10, 0}, {10, 2}, {10, 4}}

3. Discussion

It is an amazing result that such nice solutions exist.

Copyright @2004-2005 FEREGO.
All rights reserved.