Parity of the Partition Function in Arithmetic Progressions

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [23]). In this paper we prove that there indeed are infinitely many integers N in every arithmetic progression for which p(N) is even; and that there are infinitely many integers M in every arithmetic progression for which p(M) is odd so long as there is at least one (actually, we prove that if there is no such M less than 1010t7 where t is the modulus of the arithmetic progression, then p(N) must be even for all N in the arithmetic progression). Using these results we have checked Subbarao’s conjecture for all arithmetic progressions with modulus ≤ 100, 000. The main tools in our proofs are the methods developed by Deligne, Serre, and Sturm for the reduction of positive integer weight holomorphic modular forms.