On partitions of integers with restrictions involving squares

Chao Huang; Zhi-Wei Sun

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arXiv · 2105.03416

On partitions of integers with restrictions involving squares

Chao Huang·Zhi-Wei Sun

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Abstract

In this paper, we study partitions of positive integers with restrictions involving squares. We mainly establish the following two results (which were conjectured by Sun in 2013): (i) Each positive integer $n$ can be written as $n=x+y+z$ with $x,y,z$ positive integers such that $x^2+y^2+z^2$ is a square, unless $n$ has the form $n=2^{a}3^{b}$ or $2^{a}7$ with $a$ and $b$ nonnegative integers. (ii) Each integer $n>7$ with $n\not=11,14,17$ can be written as $n=x+y+2z$ with $x,y,z$ positive integers such that $x^2+y^2+2z^2$ is a square.

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Chao Huang, Zhi-Wei Sun. 2021-05-04. On partitions of integers with restrictions involving squares. https://arxiv.org/abs/2105.03416

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