On sums of four pentagonal numbers with coefficients

Dmitry Krachun; Zhi-Wei Sun; Dmitry Krachun; Zhi-Wei Sun

doi:10.3934/era.2020029

Electronic Research Archive

2020, Volume 28, Issue 1: 559-566. doi: 10.3934/era.2020029

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On sums of four pentagonal numbers with coefficients

Dmitry Krachun ^{1
,},
Zhi-Wei Sun ^{2
,
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1.
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023, St. Petersburg, Russia
2.
Department of Mathematics, Nanjing University, Nanjing 210093, China

Received: 01 January 2020
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.
- Pentagonal numbers,
- additive bases,
- ternary quadratic forms
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

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

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