On m–ary Partition Function Congruences: A Fresh Look at a Past Problem

Let bm(n) denote the number of partitions of n into powers of m. Define σr = ε2m 2 + ε3m 3 + · · · + εrm, where εi = 0 or 1 for each i. Moreover, let cr = 1 if m is odd, and cr = 2 r−1 if m is even. The main goal of this paper is to prove the congruence bm(m n− σr −m) ≡ 0 (mod m/cr). For σr = 0, the existence of such a congruence was conjectured by R. F. Churchhouse some thirty years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after. 2000 Mathematics Subject Classification: 05A17, 11P83