Parity of 4-regular and 8-regular partition functions

TL;DR

This paper characterizes when the number of 8-regular partitions of an integer is odd or even, based on a specific prime factorization condition involving the integer.

Contribution

It provides a complete characterization of the parity of 8-regular partition counts using prime factorization criteria.

Findings

Parity of $b_8(n)$ depends on the factorization $24n+7=p^{4a+1}m^2$

Established a criterion linking partition parity to prime factorization

Complete classification of parity for 8-regular partitions

Abstract

We give a complete characterization of the parity of $b_8(n)$, the number of $8$-regular partitions of $n$. Namely, we prove that $b_8(n)$ is odd or even depending on whether or not we have the factorisation $24n+7=p^{4a+1}m^2$, for some prime $p\nmid m$ and $a\ge 0$.