Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TERNARY UNIVERSAL SUMS OF GENERALIZED PENTAGONAL NUMBERS

  • Oh, Byeong-Kweon (Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University)
  • Received : 2010.04.20
  • Published : 2011.07.01
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Abstract

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

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References

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  2. Ternary Universal Sums of Generalized Polygonal Numbers pp.1793-7310, 2018, https://doi.org/10.1142/S1793042119500350