A simple proof of Watson's partition congruences for powers of 7

Abstract

Ramanujan conjectured that if n is of a specific form then p(n), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson.In this paper we establish appropriate generating formulae, from which Watson's results follow easily.Our proofs are more straightforward than those of Watson. They are elementary, depending only on classical identities of Euler and Jacobi. Watson's proofs rely on the modular equation of seventh order. We also need the modular equation but we derive it using the elementary techniques of O. Kolberg.