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 innite families of congruences modulo 5 for p_5;3(n), three innite families of congruences modulo powers of 5 for p_25;3(n), and six infnite families of congruences modulo powers of 5 for B_5;25(n). For instance, for any integers n ≥ 0 and α ≥1, we have

B_5;25(5^{2α - 1}n +5^{2α - 7}) ≡ 0 (mod 5^α).
                               6
and

B_5;25(5^{2α - 1}n +5^{2α - 7}) ≡ 0 (mod 5^α).
                               6

Key words: Partition, congruences, 2-color partition triples, (k; ℓ)-regular bipartitions.