On m-ary partition function congruences: A fresh look at a past problem

Let b(m)(n) denote the number of partitions of n into powers of m. Define s igma (r) = epsilon (2)m(2) + epsilon (3)m(3) + ... + epsilon (r)m(r), where epsilon (i) = 0 or 1 for each i. Moreover. let c(r) = 1 if m is odd, and c (r) = 2(r-1) if m is even. The main goal of this paper is to prove the cong ruence b(m)(m(r+1)n-sigma (r)-m) drop 0 (mod m(r)/c(r)). For sigma (r) = 0. the existence of such a congruence was conjectured by R. F. Churchhouse so me 30 years ago. and its truth was proved by O. J. Rodseth. G. E. Andrews. and H. Gupta soon after. (C) 2001 Academic Press.