On m-ary Partition Function Congruences: A Fresh Look at a Past Problem Original Research Article

Let bm(n) denote the number of partitions of n into powers of m. Define σr=var epsilon2m2+var epsilon3m3+…+var epsilonrmr, where var epsiloni=0 or 1 for each i. Moreover, let cr=1 if m is odd, and cr=2r−1 if m is even. The main goal of this paper is to prove the congruence bm(mr+1n−σr−m)≡0 (mod mr/cr). For σr=0, the existence of such a congruence was conjectured by R. F. Churchhouse some 30 years ago, and its truth was proved by Ø. J. Rødseth, G. E. Andrews, and H. Gupta soon after.