# 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.

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