Ramanujan-type congruences modulo powers of 5 and 7

Abstract

Let b l (n) denote the number of l-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of l. In particular, they showed that for α, n ≥ 0, b 25 (32α+3 n+2 · 32α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c N (n) counts the number of bipartitions (λ1,λ2) of n such that each part of λ2 is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, $${c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0$$ (mod 5 j+1), $${c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0$$ (mod 7 j+1) and b 25 (32α+3 · n+2 · 32α+2-1) ≡ 0 (mod 3 · 52j-1).