# Journal of the Ramanujan Mathematical Society

Volume 39, Issue 3, September 2024 pp. 281–292.

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

**Authors**:
Dazhao Tang

**Author institution:**School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R. China.

**Summary: **
In 2007, Andrews and Paule introduced the notion of broken k-diamond
partition functions. For any fixed positive integer k, let Δ{k}(n) denote
the number of broken k-diamond partitions of n. Many scholars subsequently
investigated congruence properties satisfied by Δ{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 Δ{12}(n).
Further, we conjecture that there are the corresponding congruence family and
internal congruence family modulo any powers of 5 for Δ{12}(n), which
contain the above congruences and internal congruences as special cases.

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