Affordable Access

Publisher Website

On the nonvanishing of homogeneous product sums

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
24
Issue
1
Identifiers
DOI: 10.1016/0022-314x(86)90061-2

Abstract

Abstract For integer n ≥ 1 let H n = H n ( x, y, z) = Σ p + q + r = n x p y q z r be the homogeneous product sum of weight n on three letters x, y, z. Morgan Ward conjectured that H n ≠ 0 for all integers n, x, y, z with n > 1 and xyz ≠ 0. In support of this conjecture he proved that H n ≠ 0 if n is even or if n + 2 is a prime number greater than 3. This paper adds considerably more evidence in support of Ward's conjecture by showing that in many cases H n ( a, b, c)¬=0 modulo 2, 4, or 16. The parity of H n ( a, b, c) is determined in all cases and, when H n ( a, b, c) is even, further congruences are given modulo 4 or 16.

There are no comments yet on this publication. Be the first to share your thoughts.