On the distribution of parity in the partition function

Introduction. Let p (n) be the number of (unrestricted) partitions of n, and define p (0) = 1. Then p (n) is generated byThere is little known about p (n) modulo 2; in particular, there are no known criteria for the parity of p (n) comparable in simplicity with Ramanujan's famous sufficient condition for divisibility by 5: (2) 5 | p (5fc + 4). Kolbcrg 1 proved, but by contradiction and without identifying the arguments n, that i nitely many p (n) are even, and infinitely many are odd. His proof is almost as simple as Euclid's proof that there are infinitely many primes, but like that proof it offers only very little more in the way of exact information concerning questions of distribution. From Gupta's tables 2, 3 we find the following cumulative distribution into odds and evens for 0 ^ n: £ 499.