Congruence properties modulo powers of \(2\) for \(4\)-regular partitions (Q6614187)

This paper provides a valuable and insightful contribution to the study of congruence properties of \(\ell\)-regular partitions, specifically focusing on \(b_4(n)\), the number of \(4\)-regular partitions of \(n\). The authors successfully build upon the work of notable researchers like Andrews, Hirschhorn, Sellers [\textit{G. E. Andrews} et al., Ramanujan J. 23, No. 1--3, 169--181 (2010; Zbl 1218.05018)], and others, presenting a fresh and innovative approach to proving \textit{self-similar} congruence properties of the generating function for \(b_4(n)\). This approach is both elegant and powerful, as it leads to an impressive array of congruence families modulo powers of \(2\).\N\NOne of the key strengths of the paper is its ability to not only generalize previous results but also complement the comprehensive study by \textit{W. J. Keith} and \textit{F. Zanello} [J. Number Theory 235, 275--304 (2022; Zbl 1491.11096)] on \(\ell\)-regular partition functions. The authors demonstrate a deep understanding of the subject matter, extending known results while introducing new ideas that open doors for further research. Their conjectures on congruence families, internal congruence families, and self-similar congruences for \(4\)-, \(8\)-, and \(16\)-regular partitions are thought-provoking and present exciting challenges for future exploration.\N\NOverall, this paper offers a well-rounded and significant contribution to the field of partition theory. It enriches the existing literature on congruences for regular partitions, providing both a fresh perspective and a strong foundation for subsequent studies.