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Some variants of Lagrange's four squares theorem

Volume 183 / 2018

Yu-Chen Sun, Zhi-Wei Sun Acta Arithmetica 183 (2018), 339-356 MSC: Primary 11E25; Secondary 11D85, 11E20, 11P05. DOI: 10.4064/aa170508-14-3 Published online: 25 May 2018

Abstract

Lagrange’s four squares theorem is a classical result in number theory. Recently, Z.-W. Sun found that it can be further refined in various ways. In this paper we study some conjectures of Sun and obtain various refinements of Lagrange’s theorem. We show that any nonnegative integer can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb Z)$ with $x+y+z+w$ (or $x+y+z+2w$, or $x+2y+3z+w$) a square (or a cube). Also, every $n=0,1,2,\ldots$ can be represented by $x^2+y^2+z^2+w^2$ $(x,y,z,w\in{\mathbb Z})$ with $x+y+3z$ (or $x+2y+3z$) a square (or a cube), and each $n=0,1,2,\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in{\mathbb Z})$ with $(10w+5x)^2+(12y+36z)^2$ (or $x^2y^2+9y^2z^2+9z^2x^2$) a square. We also provide an advance on the 1-3-5 conjecture of Sun. Our main results are proved by a new approach involving Euler’s four-square identity.

Authors

  • Yu-Chen SunDepartment of Mathematics
    Nanjing University
    Nanjing 210093, People’s Republic of China
    e-mail
  • Zhi-Wei SunDepartment of Mathematics
    Nanjing University
    Nanjing 210093, People’s Republic of China
    e-mail

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