TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS
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For an integer m ≥ 3, every integer of the form pm(x) = <수식> with x ∈ Z is said to be a generalized m-gonal number. Let <수식> 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 = apk(x)+bpk(y)+cpk(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.
Abstract
For an integer m ≥ 3, every integer of the form pm(x) = <수식> with x ∈ Z is said to be a generalized m-gonal number. Let <수식> 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 = apk(x)+bpk(y)+cpk(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.
참고문헌 (16)
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