New parity results for 3-regular partitions

Abstract

Let b_t(n) count the number of t-regular partitions of n. In recent years, congruence properties of b_t(n) have received a lot of attention and a number of infiinite families of congruences modulo 2 for b_t(n) have been proved for some small t. Very recently, Keith and Zanello investigated the parity of the coefficients of certain eta-quotients and proved new parity results for b_t(n). In particular, they discovered new infinite families of congruences modulo 2 for b₃(n) and those congruences involve every prime p with p ≡ 13; 17; 19; 23 (mod 24). Motivated by their work, we prove new infinite families of congruences modulo 2 for b₃(n) and these congruences involve every prime p ≥ 5 based on Newman's results. Additionally, we proved new congruences for b₃(n) which implies that there exist infinitely many positive integers m such that b₃(m) is odd.