Parity of the Partition Function

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 [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t when t = 1, 2, 3, 4, 5, 10, 12, 16, and 40. Here we announce 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 such M . In fact if there is such an M , then the smallest such M ≤ 1010t7. Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus t ≤ 100, 000.