TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS

Abstract

For an integer <TEX>$m{\geq}3$</TEX>, every integer of the form <TEX>$p_m(x)$</TEX> = <TEX>$\frac{(m-2)x^2(m-4)x}{2}$</TEX> with x <TEX>${\in}$</TEX> <TEX>$\mathbb{Z}$</TEX> is said to be a generalized m-gonal number. Let <TEX>$a{\leq}b{\leq}c$</TEX> 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 = <TEX>$ap_k(x)+bp_k(y)+cp_k(z)$</TEX>. Sun proved in [16] that, when k = 5 or <TEX>$k{\geq}7$</TEX>, 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.