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- W2266234140 abstract "Let $p(n)$ denote the number of partitions of a non-negative integer $n$. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers $M$ for which $p(M)$ is odd, as well as infinitely many integers $N$ for which $p(N)$ is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus $t$ when $t=1,2,3,4,5,10,12,16,$ and $40.$ Here we announce that there indeed are infinitely many integers $N$ in every arithmetic progression for which $p(N)$ is even; and that there are infinitely 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 $Mleq 10^{10}t^7$. Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus $tleq 100,000$." @default.
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- W2266234140 doi "https://doi.org/10.1090/s1079-6762-95-01005-5" @default.
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