Journal of the Ramanujan Mathematical Society

Volume 40, Issue 3, September 2025 pp. 245–256.

An iterative approach for proving congruences of partition k-tuples with t-cores

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

Summary: Let A{t,k}(n) denote the number of partition k-tuples of n with t-cores. In this paper, we present an iterative approach for guessing and proving congruences of A{t,ai+b}(n) by utilizing three recurrences based on the modular equations of fifth, seventh and thirteenth orders, where t ∈ {5,7,13} and ai+b is some arithmetic progression. In particular, we prove ten conjectural congruences modulo powers of 5 satisfied by A{5,ai+b}(n), which were recently posed by Saikia, Sarma and Talukdar (Indian J.~Pure Appl. Math., 2024). Further, we pose a conjecture on congruences modulo any powers of t for A{t,ai+b}(n).

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