TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS

For an integer m >= 3, every integer of the form p(m)(x) = (m-2)x(2)-(m-4)x/2 with x is an element of Z is said to be a generalized m-gonal number. Let a <= b <= 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 = ap(k)(x) + b(pk)(y) + cp(k)(z). Sun proved in [16] that, when k = 5 or k >= 7, there are only 20 candidates for universal quadruples, which he listed explicitly and which all involve only the case of pentagonal numbers (k = 5). He verified 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.