Arithmetic properties for two restricted partitions modulo powers of 5

Abstract

Let p_k,3(n) count the number of 2-color partition triples of n where one of the colors appears only in parts that are multiples of k and B_k,ℓ(n) denote the number of (k, ℓ)-regular bipartitions of n. In this paper, we prove two infinite families of congruences modulo 5 for p_5,3(n), three infinite families of congruences modulo powers of 5 for p_25,3(n), and six infinite families of congruences modulo powers of 5 for B_5,25(n). For instance, for any integers n ≥ 0 and α ≥ 1, we have