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

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(n−ω(k))+p(n−ω(−k))),                                        (*) where ω(k)=(3k 2+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 i≥1, finite sets Mi for which p(n) is odd for an odd number of n∈Mi.