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 + · · · + εrmr, 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 r+1n − σr −m) ≡ 0 (mod mr/cr). For σr = 0, the existence of such a congruence was conjectured by R. F. Church-house some thirty years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after