Ramamurthi fractions permutation results

by FEREGO Research

1. Introduction

Define the linear fraction form 

     P(m) - Q(m)
  Z({P,Q,R,S},m) = -------------
     R(m) - S(m)
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.

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 

  Z({P,Q,R,S},4) = Z({P,Q,R,S},2)
and in short hand form this is called the identity for {4, 2} for the permutation {P, Q, R, S}. Previous studies have documented some results for the permutation {P, Q, R, S}.

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

  Z(P*,m) = Z(P*,n) .
Note that the given permutation P* is the same on both sides of the equation.

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 permutation.

2. Results

2.1. Solutions for {P, Q, R, S}

There are 7 solutions for {P, Q, R, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {7, 3} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {7, 5} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.2. Solutions for {P, Q, S, R}

There are 7 solutions for {P, Q, S, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {7, 3} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {7, 5} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.3. Solutions for {P, R, Q, S}

There are 7 solutions for {P, R, Q, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

  11
-(--)
  3

The {7, 3} solution is 

  11
-(--)
  3

The {7, 5} solution is 

  11
-(--)
  3

The {8, 6} solution is 

  34
-(--)
  57

The {10, 6} solution is 

  34
-(--)
  57

2.4. Solutions for {P, R, S, Q}

There are 7 solutions for {P, R, S, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

11
--
3

The {7, 3} solution is 

11
--
3

The {7, 5} solution is 

11
--
3

The {8, 6} solution is 

34
--
57

The {10, 6} solution is 

34
--
57

2.5. Solutions for {P, S, Q, R}

There are 7 solutions for {P, S, Q, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.6. Solutions for {P, S, R, Q}

There are 7 solutions for {P, S, R, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 3} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 5} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.7. Solutions for {Q, P, R, S}

There are 7 solutions for {Q, P, R, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {7, 3} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {7, 5} solution is 

      3    2           2      3
  -6 a  + a  b + 19 a b  + 6 b
-(-----------------------------)
     3       2        2      3
  6 a  + 19 a  b + a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.8. Solutions for {Q, P, S, R}

There are 7 solutions for {Q, P, S, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {7, 3} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {7, 5} solution is 

    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b
-----------------------------
   3       2        2      3
6 a  + 19 a  b + a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.9. Solutions for {Q, R, P, S}

There are 7 solutions for {Q, R, P, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.10. Solutions for {Q, R, S, P}

There are 7 solutions for {Q, R, S, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.11. Solutions for {Q, S, P, R}

There are 7 solutions for {Q, S, P, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

  3
-(--)
  11

The {7, 3} solution is 

  3
-(--)
  11

The {7, 5} solution is 

  3
-(--)
  11

The {8, 6} solution is 

  57
-(--)
  34

The {10, 6} solution is 

  57
-(--)
  34

2.12. Solutions for {Q, S, R, P}

There are 7 solutions for {Q, S, R, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

3
--
11

The {7, 3} solution is 

3
--
11

The {7, 5} solution is 

3
--
11

The {8, 6} solution is 

57
--
34

The {10, 6} solution is 

57
--
34

2.13. Solutions for {R, P, Q, S}

There are 7 solutions for {R, P, Q, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

11
--
3

The {7, 3} solution is 

11
--
3

The {7, 5} solution is 

11
--
3

The {8, 6} solution is 

34
--
57

The {10, 6} solution is 

34
--
57

2.14. Solutions for {R, P, S, Q}

There are 7 solutions for {R, P, S, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

  11
-(--)
  3

The {7, 3} solution is 

  11
-(--)
  3

The {7, 5} solution is 

  11
-(--)
  3

The {8, 6} solution is 

  34
-(--)
  57

The {10, 6} solution is 

  34
-(--)
  57

2.15. Solutions for {R, Q, P, S}

There are 7 solutions for {R, Q, P, S}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
----------------------------------
     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.16. Solutions for {R, Q, S, P}

There are 7 solutions for {R, Q, S, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.17. Solutions for {R, S, P, Q}

There are 7 solutions for {R, S, P, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {7, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {7, 5} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.18. Solutions for {R, S, Q, P}

There are 7 solutions for {R, S, Q, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {7, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {7, 5} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.19. Solutions for {S, P, Q, R}

There are 7 solutions for {S, P, Q, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 3} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 5} solution is 

     3       2           2       3
-12 a  + 17 a  b + 53 a b  + 12 b
----------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.20. Solutions for {S, P, R, Q}

There are 7 solutions for {S, P, R, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 3} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {7, 5} solution is 

    3       2           2       3
12 a  - 17 a  b - 53 a b  - 12 b
---------------------------------
    3       2           2       3
12 a  + 53 a  b + 17 a b  - 12 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.21. Solutions for {S, Q, P, R}

There are 7 solutions for {S, Q, P, R}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

3
--
11

The {7, 3} solution is 

3
--
11

The {7, 5} solution is 

3
--
11

The {8, 6} solution is 

57
--
34

The {10, 6} solution is 

57
--
34

2.22. Solutions for {S, Q, R, P}

There are 7 solutions for {S, Q, R, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

  3
-(--)
  11

The {7, 3} solution is 

  3
-(--)
  11

The {7, 5} solution is 

  3
-(--)
  11

The {8, 6} solution is 

  57
-(--)
  34

The {10, 6} solution is 

  57
-(--)
  34

2.23. Solutions for {S, R, P, Q}

There are 7 solutions for {S, R, P, Q}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {7, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {7, 5} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
----------------------------
   3    2           2      3
6 a  - a  b - 19 a b  - 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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

2.24. Solutions for {S, R, Q, P}

There are 7 solutions for {S, R, Q, P}

The solutions are: {{4, 2}, {5, 3}, {7, 3}, {7, 5}, {8, 6}, {10, 6}, {10, 8}}

The {4, 2} solution is 

0

The {5, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {7, 3} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {7, 5} solution is 

   3       2        2      3
6 a  + 19 a  b + a b  - 6 b
-----------------------------
    3    2           2      3
-6 a  + a  b + 19 a b  + 6 b

The {8, 6} solution is 

     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

The {10, 6} solution is 

     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. Discussion

It is an amazing result that such nice solutions exist. We have found that each permutation of the set {P, Q, R, S} has seven solutions. The seven solutions {m, n} for each permutation are the same seven indices; this is an interesting result. But we find the solutions is not completely symmetric in that the algebraic value of the identities can be different rational functions of a and b. Thus the solutions are not all identical.

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All rights reserved.