New parity results of sums of partitions and squares in arithmetic progressions

Abstract

Recently, Ballantine and Merca proved that if $ (a,b) \in \{(6,8), (8,12), (12,24), (15,40), (16,48), (20,120), (21,168)\}$, then $\sum_{ak+1 {\rm square}}p(n-k)\equiv 1 ({\rm mod} 2)$ if and only if $bn+1$ is a square. In this paper, we investigate septuple $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)\in \mathbb{N}^5\times \mathbb{Q}^2$ for which $\sum_{a_1k+a_2 {\rm square}}p(a_3a_4^\alpha n+a_6 a_4^\alpha+a_7-k) \equiv 1 ({\rm mod} 2)$ if and only if $a_5n+1$ is a square. We prove some new parity results of sums of partitions and squares in arithmetic progressions which are analogous to the results due to Ballantine and Merca.