### Electronic Research Archive

2020, Issue 1: 559-566. doi: 10.3934/era.2020029
Special Issues

# On sums of four pentagonal numbers with coefficients

• Primary: 11B13, 11E25; Secondary: 11D85, 11E20, 11P70

• 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 integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat.

Citation: Dmitry Krachun, Zhi-Wei Sun. On sums of four pentagonal numbers with coefficients[J]. Electronic Research Archive, 2020, 28(1): 559-566. doi: 10.3934/era.2020029

### Related Papers:

• 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 integeris a sum of five pentagonal numbers two of which are equal; this refines a classical resultof Cauchy claimed by Fermat. ###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142 0.8 0.9

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