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Parity of the partition function

Author: Ken Ono
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 35-42
MSC (1991): Primary 05A17; Secondary 11P83
MathSciNet review: 1336698
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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 $M\leq 10^{10}t^7$. Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus $t\leq 100,000$.

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Additional Information

Ken Ono
Affiliation: address Department of Mathematics, The University of Illinois, Urbana, Illinois 61801

Keywords: Parity conjecture, partitions, modular forms
Received by editor(s): February 28, 1995
Received by editor(s) in revised form: May 3, 1995
Additional Notes: The author is supported by NSF grant DMS-9508976.
Communicated by: Don Zagier
Article copyright: © Copyright 1995 American Mathematical Society

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