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On partitions of integers with restrictions involving squares (2105.03416v2)

Published 4 May 2021 in math.NT

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 $x2+y2+z2$ 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 $x2+y2+2z2$ is a square.

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