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

DOI:
https://doi.org/10.1090/S1079-6762-95-01005-5

MathSciNet review:
1336698

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the number of partitions of a non-negative integer . A well-known conjecture asserts that every arithmetic progression contains infinitely many integers for which is odd, as well as infinitely many integers for which is even (see Subbarao [ 22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus when and Here we announce that there indeed are infinitely many integers in every arithmetic progression for which is even; and that there are infinitely many integers in every arithmetic progression for which is odd so long as there is at least one such . In fact if there is such an , then the smallest such . Using these results and a fair bit of machine computation, we have verified the conjecture for every arithmetic progression with modulus .

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Additional Information

**Ken Ono**

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

Email:
ono@symcom.math.uiuc.edu

DOI:
https://doi.org/10.1090/S1079-6762-95-01005-5

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