Some Ramanujan Number 1729 Studies

results by FEREGO Research and others

Ramanujan is known for some marvelous, beautiful, and incredibly obscure results in number theory. Whereas many results in mathematics follow intuition Ramanujan's results just seem to “come out of a hat” and are some of the most non-intuitive and obscure results in all of mathematics. Ramanujan would find results that left the rest of the mathematical world shocked, stunned, amazed, and even somewhat envious. He is truly one of the deep geniuses of mathematics.

We have done some work regarding the work found at this location, which we are citing in particular as a reference. Please read the amazing results at this referenced location, the interpretation given, as well as various expansions. The fraction expansion equivalents of these results are amazing results!

We have verified some of the results given at the above reference, and have been attempting to do further work. We have tried to double check some of his results, and are working on finding more results and generalizations, if possible.

We think these are really amazing results. Also it appears (for various reasons) that these results (with respect to the specified reference above) are new results, and were (apparently) not known to Ramanujan, and are not apparent generalizations of Ramanujan's so-called “third notebook” formula. We do not wish to be misleading: we are not experts on Ramanujan.

The functions P, Q, R, and S are as defined at the reference above, so we do not repeat the definitions and assumptions. The specified values of a and b are also as given there.

Multiplying out the fractions, and expressing the results as done at the above reference, reveals some amazing results.

( P(m)P(n) - Q(m)Q(n) ) / ( R(m)R(n) - S(m)S(n) ) = 51/23

We have verified the results {3,5}, {5,7} and {7,3} which are given at the above reference.

We have found this result to be true for more values. In particular, we have found some additional pairs {m,n} for which this is true.

( P(m) / P(n) - Q(m) / Q(n) ) / ( R(m) / R(n) - S(m) / S(n) ) = 51/23

We have verified the results {9,3}, {11,5} and {13,7} which are given at the above reference.

We have found this result to be true for more values. In particular, we have found some additional pairs {m,n} for which this is true.

More. We are trying to create the fraction expansions of these results, as given at the above reference.

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Some Ramanujan Number 1729 Studies

( P(m)*P(n) - Q(m)*Q(n) ) / ( R(m)*R(n) - S(m)*S(n) ) = 51/23

( P(m) / P(n) - Q(m) / Q(n) ) / ( R(m) / R(n) - S(m) / S(n) ) = 51/23

( P(m)P(n) - Q(m)Q(n) ) / ( R(m)R(n) - S(m)S(n) ) = 51/23