TY - JOUR

T1 - On Sums Of Four Pentagonal Numbers With Coefficients

AU - Krachun, Dmitry

AU - Sun, Zhi Wei

N1 - Funding Information:
Received by the editors January, 2020. 2010 Mathematics Subject Classification. Primary: 11B13, 11E25; Secondary: 11D85, 11E20, 11P70. Key words and phrases. Pentagonal numbers, additive bases, ternary quadratic forms. The work is supported by the NSFC(Natural Science Foundation of China)-RFBR(Russian Foundation for Basic Research) Cooperation and Exchange Program (grants NSFC 11811530072 and RFBR 18-51-53020-GFEN-a). The second author is also supported by the Natural Science Foundation of China (grant no. 11971222). ∗ Corresponding author: Zhi-Wei Sun.
Publisher Copyright:
© 2020. American Institute of Mathematical Sciences

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 - additive bases

KW - Pentagonal numbers

KW - ternary quadratic forms.

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U2 - 10.3934/era.2020029

DO - 10.3934/era.2020029

M3 - Article

AN - SCOPUS:85098439093

SN - 1935-9179

VL - 28

SP - 559

EP - 566

JO - Electronic Research Archive

JF - Electronic Research Archive

IS - 1

ER -