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arXiv · 2001.05325

On sums of four pentagonal numbers with coefficients

Dmitry Krachun·Zhi-Wei Sun
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Abstract

The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$ can be written as $w+bx+cy+dz$ with $w,x,y,z$ pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.

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Dmitry Krachun, Zhi-Wei Sun. 2020-01-14. On sums of four pentagonal numbers with coefficients. https://arxiv.org/abs/2001.05325

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