On Sums Of Four Pentagonal Numbers With Coefficients

Dmitry Krachun; Zhi Wei Sun

doi:10.3934/era.2020029

On Sums Of Four Pentagonal Numbers With Coefficients

Dmitry Krachun, Zhi Wei Sun

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

The pentagonal numbers are the integers given by p5(n) = n(3n-1)/2 (n = 0, 1, 2,…). 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,… 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.

Original language	English (US)
Pages (from-to)	559-566
Number of pages	8
Journal	Electronic Research Archive
Volume	28
Issue number	1
DOIs	https://doi.org/10.3934/era.2020029
State	Published - Mar 2020
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

Pentagonal numbers
additive bases
ternary quadratic forms.

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10.3934/era.2020029

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title = "On Sums Of Four Pentagonal Numbers With Coefficients",

abstract = "The pentagonal numbers are the integers given by p5(n) = n(3n-1)/2 (n = 0, 1, 2,…). 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,… 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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author = "Dmitry Krachun and Sun, {Zhi Wei}",

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year = "2020",

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doi = "10.3934/era.2020029",

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T1 - On Sums Of Four Pentagonal Numbers With Coefficients

AU - Krachun, Dmitry

AU - Sun, Zhi Wei

PY - 2020/3

Y1 - 2020/3

N2 - The pentagonal numbers are the integers given by p5(n) = n(3n-1)/2 (n = 0, 1, 2,…). 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,… 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.

AB - The pentagonal numbers are the integers given by p5(n) = n(3n-1)/2 (n = 0, 1, 2,…). 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,… 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.

KW - Pentagonal numbers

KW - additive bases

KW - ternary quadratic forms.

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ER -

On Sums Of Four Pentagonal Numbers With Coefficients

Abstract

All Science Journal Classification (ASJC) codes

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