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.
We prove the conjecture for every arithmetic progression whose modulus is a power of 2.