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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 $t\leq 100,000$.

Copyright American Mathematical Society 1995

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Article Info

• ERA Amer. Math. Soc. 01 (1995), pp. 35-42
• Publisher Identifier: S 1079-6762(95)01005-5
• 1991 Mathematics Subject Classification. Primary 05A17; Secondary 11P83 .
• Key words and phrases. Parity conjecture, partitions, modular forms
• Received by the editors February 28, 1995, and, in revised form, May 3, 1995
• Communicated by Don Zagier
• Comments (When Available)

Ken Ono

Department of Mathematics, The University of Illinois, Urbana, Illinois 61801

E-mail address: ono@symcom.math.uiuc.edu

The author is supported by NSF grant DMS-9508976.