A Proof of Subbarao’s Conjecture

Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) for which p(N) is even, and infinitely many integers M ≡ r (mod t) for which p(M) is odd. In the even case the conjecture was settled by Ken Ono. In this paper we prove the odd part of the conjecture which together with Ono’s result implies the full conjecture. We also prove that for every arithmetic progression r (mod t) there are infinitely many integers N ≡ r (mod t) such that p(N) 6≡ 0 (mod 3), which settles an open problem posed by Scott Ahlgren and Ken Ono.