Q-series, elliptic curves, and odd values of the partition function. (1st March 1999) - Eriksson, Nicholas
Abstract : Letp ( n ) be the number of partitions of an integern . Euler proved the following recurrence forp ( n ) :p ( n ) = ∑ k = 1 ∞ ( − 1 ) k + 1 ( p ( n − ω ( k ) ) + p ( n − ω ( − k ) ) ), ( * ) whereω ( k ) = ( 3 k 2 + k ) / 2 . In view of Euler's result, one sees that it is fairly easy to computep ( n ) very quickly. However, many questions remain open even regarding the parity ofp ( n ) . In this paper, we use various facts about elliptic curves andq -series to construct, for everyi ≥ 1, finite setsM i for whichp ( n ) is odd for an odd number ofn ∈ M i .