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International Journal of Mathematics and Mathematical Sciences
Volume 22, Issue 1, Pages 55-65
http://dx.doi.org/10.1155/S0161171299220558

q-series, elliptic curves, and odd values of the partition function

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Received 28 January 1997

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let p(n) be the number of partitions of an integer n. Euler proved the following recurrence for p(n): p(n)=k=1(1)k+1(p(nω(k))+p(nω(k))),(*) where ω(k)=(3k2+k)/2. In view of Euler's result, one sees that it is fairly easy to compute p(n) very quickly. However, many questions remain open even regarding the parity of p(n). In this paper, we use various facts about elliptic curves and q-series to construct, for every i1, finite sets Mi for which p(n) is odd for an odd number of nMi.