Refining Lagrange's four-square theorem

@article{Sun2016RefiningLF,
  title={Refining Lagrange's four-square theorem},
  author={Zhi-Wei Sun},
  journal={arXiv: Number Theory},
  year={2016}
}
  • Zhi-Wei Sun
  • Published 22 April 2016
  • Mathematics
  • arXiv: Number Theory
Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the… Expand
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