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 H...