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Parity of the Partition Function
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## Parity of the partition function

### Ken Ono

**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 $t\leq 100,000$.

*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.

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