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Representing \({\varvec{n}}\) as \({\varvec{n=x+y+z}}\) with \({\varvec{x^2+y^2+z^2}}\) a square

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Abstract

In this paper, we mainly confirm the following conjecture of Sun posed in 2013: 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.

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Acknowledgements

We are indebted to the referee for helpful comments.

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Authors and Affiliations

  1. School of Mathematics and Physics, Anqing Normal University, Anqing, 246133, People’s Republic of China

    Chao Huang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Zhi-Wei Sun

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  1. Chao Huang
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  2. Zhi-Wei Sun
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Correspondence to Zhi-Wei Sun.

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Supported by the Natural Science Foundation of China (grant no. 11971222).

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Huang, C., Sun, ZW. Representing \({\varvec{n}}\) as \({\varvec{n=x+y+z}}\) with \({\varvec{x^2+y^2+z^2}}\) a square. Arch. Math. 121, 231–239 (2023). https://doi.org/10.1007/s00013-023-01896-5

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  • DOI: https://doi.org/10.1007/s00013-023-01896-5

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