Parity of the Partition Functionken

Let p(n) denote the number of partitions of a non-negativeinteger n. A well-known conjecture asserts that every arithmetic progression contains innnitely many integers M for which p(M) is odd, as well as innnitely many integers N for which p(N) is even (see Subbarao 22]). From the works of various authors, this conjecture has been veriied for every arithmetic progression with modulus t when t Here we announce that there indeed are innnitely many integers N in every arithmetic progression for which p(N) is even; and that there are innnitely 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 10 10 t 7. Using these results and a fair bit of machine computation, we have veriied the conjecture for every arithmetic progression with modulus t 100;000.