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

Abstract

For a positive integer $t\ge 2$, 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 $t\le 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13,17,19,23(\mathrm{mod}24)$. In this paper, we investigate the parity of $b_{21}(n)$ involving the primes $p$ with $p\equiv 1,5,7,11(\mathrm{mod}24)$. 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 $j\in \left\{1,4,8\right\}$, 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 $p\ge 5$ 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.

15 pages