Schur’s 1926 partition theorem and related identities

Abstract

In this paper we study Schur’s 1926 Partition Theorem rigorously and in depth. The fun damental partition theorem by Schur asserts the equality of the number of partitions of n into parts congruent to ±1 (mod 6) and the number of partitions of n into parts that differ by at least 3 with multiples of 3 differing by at least 6. We aim to throw more light on the proof of the aforementioned theorem from its origins through to its generalizations. Conse quently, from our desire to tackle the theorem from its foundation, we also examine related identities that led to the discovery of the theorem which includes Euler’s Theorem together with its extensions, Euler Pairs and Glaisher’s Theorem. We will especially showcase two different proofs of the best known generalization of Schur’s Theorem, and make the proofs more comprehensive