Certain Diophantine equations and new parity results for $21$-regular partitions

Abstract

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13, 17, 19, 23 \pmod{24}$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1, 5, 7, 11 \pmod{24}$. We prove new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving the odd primes $p$ for which the Diophantine equation $8x^2+27y^2=jp$ has primitive solutions for some $j\in\left\lbrace1,4,8\right\rbrace$, and we also prove that the Dirichlet density of such primes is equal to $1/6$. Recently, Yao provided new infinite families of congruences modulo $2$ for $b_{3}(n)$ and those congruences involve every prime $p\geq 5$ based on Newman's results. Following a similar approach, we prove new infinite families of congruences modulo $2$ for $b_{21}(n)$, and these congruences imply that $b_{21}(n)$ is odd infinitely often.