Let bm(n) denote the number of partitions of n into powers of m. Define σr=ε2m2+ε3m3+...+εrmr, where εi=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 (modmr/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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory