Congruence properties modulo powers of 5 for broken 12-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partition functions. For any fixed positive integer k, let triangle(k)(n)denote the number of broken k-diamond partitions of n. Many scholars subsequently investigated congruence properties satisfied by triangle(k)(n)with different moduli. However, there are a few results on congruences modulo powers of 5 for this partition function family. In this paper, we prove three congruences and three internal congruences modulo small powers of 5 for triangle(12)(n). Further, we conjecture that there are the corresponding congruence family and internal congruence family modulo any powers of 5 for triangle(12)(n), which contain the above congruences and internal congruences as special cases.