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

q-SERIES, ELLIPTIC CURVES, AND ODD VALUES OF THE PARTITION FUNCTION

Letp(n) be the number of partitions of an integern. Euler proved the following recurrence for p(n): p(n)= ∞ ∑ k=1 (−1)k+1 ( p ( n−ω(k))+p(n−ω(−k))), (∗) whereω(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 ...

Odd values of the partition function

Let p(n) denote the number of partitions of an integer n. Recently the author has shown that in any arithmetic progression r (mod t), there exist infinitely many N for which p(N) is even, and there are infinitely many M for which p(M) is odd, provided there is at least one such M. Here we construct finite sets of integers Mi for which p(Mi) is odd for an odd number of i. Whereas Euler’s recurre...

Elliptic Curves over Q

All polynomials and rational functions in this essay are assumed to have coefficients in Q . Fix an integer n ≥ 1. An affine variety is a simultaneous irreducible system of polynomial equations in n variables. The Q -points, R -points and C points of the affine variety are all solutions of the polynomial system in Q , R n and C , respectively. Rational projective n-spaceg Q n is the set of line...

Elliptic Curves over Q

ELLIPTIC CURVES OVER Q(i)

A study of the diophantine equation v2 = 2u4 − 1 led the authors to consider elliptic curves specifically over Q(i) and to examine the parallels and differences with the classical theory over Q. In this paper we present some extensions of the classical theory along with some examples illustrating the results. The well-known diophantine equation v2 = 2u4 − 1 , has, ignoring signs, only two integ...