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

Nicholas Eriksson

International Journal of Mathematics and Mathematical Sciences, 1999, vol. 22, 1-11

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 ) = ( 3 k 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 M i for which p ( n ) is odd for an odd number of n ∈ M i .

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:796128

DOI: 10.1155/S0161171299220558

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