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

摘要

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 $sum_{ak+1 {m square}}p(n-k)equiv 1 ({m mod} 2)$ if and only if $bn+1$ is a square. In this paper, we investigate septuple $(a_1,a_2,a_3,a_4,a_5,a_6,a_7)in mathbb{N}^5imes mathbb{Q}^2$ for which $sum_{a_1k+a_2 {m square}}p(a_3a_4^lpha n+a_6 a_4^lpha+a_7-k) equiv 1 ({m mod} 2)$ 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.