New parity results of sums of partitions and squares in arithmetic progressions
DOI:
https://doi.org/10.11575/cdm.v14i1.62644Keywords:
Number TheoryAbstract
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.
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