THE DETERMINATION OF RAMANUJAN PAIRS.

Abstract

We call two increasing sequences of positive integers {aᵢ}, {b(j)} a "Ramanujan Pair" if the following identity holds: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). The goal of this investigation is to determine all Ramanujan Pairs. Although this goal was not completely reached, we have determined all pairs for which the first term a₁ ≥ 5 and have proved that any Ramanujan Pair which begins with a₁ = m, where 1 ≤ m ≤ 4, aside from the known pairs, would have to branch off the first Euler identity with {aᵢ} = {i + m - 1}, {b(j)} = {j m}. A great deal of computing was done to discover the proofs given here. The search methods used and their programs are discussed in detail. Beyond these results, we have found all finite Ramanujan Pairs. Finally, modular Ramanujan Pairs (where the coefficients in the identity are reduced modulo n) are also examined.