Congruences modulo powers of 5 for partition <i>k</i>-tuples with 5-cores

Abstract A partition is called a t -core if none of its hook lengths is a multiple of t . Let $a_t(n)$ denote the number of t -core partitions of n . Garvan, Kim and Stanton proved that for any $n\geq1$ and $m\geq1$ , $a_t\big(t^mn-(t^2-1)/24\big)\equiv0\pmod{t^m}$ , where $t\in\{5,7,11\}$ . Let $A_{t,k}(n)$ denote the number of partition k -tuples of n with t -cores. Several scholars have been subsequently investigated congruence properties modulo high powers of 5 for $A_{5,k}(n)$ with $k\in\{2,3,4\}$ . In this paper, by utilizing a recurrence related to the modular equation of fifth order, we establish dozens of congruence families modulo high powers of 5 satisfied by $A_{5,k}(n)$ , where $4\leq k\leq25$ . Moreover, we deduce an infinite family of internal congruences modulo high powers of 5 for $A_{5,4}(n)$ . In particular, we generalize greatly a recent result on a congruence family modulo high powers of 5 enjoyed by $A_{5,4}(n)$ , which was proved by Saikia, Sarma and Talukdar (Indian J. Pure Appl. Math., 2024). Finally, we conjecture that there exists a similar phenomenon for $A_{5,k}(n)$ with $k\geq26$ .