The 2-adic behavior of the number of partitions into distinct parts

摘要

Let Q(n) denote the number of partitions of an integer n into distinct parts. For positive integers j, the first author and B. Gordon proved that Q(n) is a multiple of 2(j) for every non-negative integer n outside a set with density zero. Here we show that if i not equivalent to 0 (mod 2(j)), then #{0 less than or equal to n less than or equal to X : Q(n) equivalent to i (mod 2(f))} much greater than (j) rootX/log X. In particular, Q(n) lies in every residue class modulo 2(j) infinitely often. In addition, we examine the behavior of Q(n) (mod 8) in detail, and we obtain a simple "closed formula" using the arithmetic of the ring Z[root -6]. (C) 2000 Academic Press. [References: 19]