NEW PARITY RESULTS OF SUMS OF PARTITIONS AND SQUARES IN ARITHMETIC PROGRESSIONS

Recently, Ballantine and Merca proved that if (a, b) is an element of {(6, 8), (8,12), (12,24), (15,40), (16,48), (20,120), (21,168)}, then Sigma(ak+1 square) p(n - k) 1 (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)) is an element of N-5 x Q(2) for which Sigma(a1k+a2 square) P(a(3)a(4)(alpha)n + a(6)a(4)(alpha) + a(7) - k) 1 (mod 2) if and only if a(5)n + 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.