TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS

Title & Authors
TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS
Oh, Byeong-Kweon;

Abstract
For an integer $\small{m{\geq}3}$, every integer of the form $\small{p_m(x)}$ = $\small{\frac{(m-2)x^2(m-4)x}{2}}$ with x $\small{{\in}}$ $\small{\mathbb{Z}}$ is said to be a generalized m-gonal number. Let $\small{a{\leq}b{\leq}c}$ and k be positive integers. The quadruple (k, a, b, c) is said to be universal if for every nonnegative integer n there exist integers x, y, z such that n = $\small{ap_k(x)+bp_k(y)+cp_k(z)}$. Sun proved in [16] that, when k = 5 or $\small{k{\geq}7}$, there are only 20 candidates for universal quadruples, which h listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He veri ed that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only (5, 1, 1, t) for t = 6, 8, 9, 10, (5, 1, 2, 8) and (5, 1, 3, s) for s = 7, 8 as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.
Keywords
generalized polygonal numbers;ternary universal sums;
Language
English
Cited by
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