Subbarao’s Conjecture on the Parity of the Partition Function

Let p(n) denote the ordinary partition function. In 1966, Subbarao [18] conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N (resp. M) ≡ r (mod t) for which p(N) is even (resp. odd). We prove Subbarao’s conjecture for all moduli t of the form m · 2 where m ∈ {1, 5, 7, 17}. To obtain this theorem we make use of recent results of Ono and Taguchi [14] on the nilpotent action of Hecke algebras on certain spaces of modular forms modulo 2.