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Sums of integers and sums of their squares

Volume 194 / 2020

Detlev W. Hoffmann Acta Arithmetica 194 (2020), 295-313 MSC: Primary 11E25; Secondary 11D04, 11D09. DOI: 10.4064/aa190219-15-10 Published online: 13 March 2020

Abstract

Suppose a positive integer $n$ is written as a sum of squares of $m$ integers. What can one say about the value $T$ of the sum of these $m$ integers itself? Which $T$ can be obtained if one considers all possible representations of $n$ as a sum of squares of $m$ integers? Denote this set of all possible $T$ by ${\mathscr S}_m(n)$. Goldmakher and Pollack have given a simple characterization of ${\mathscr S}_4(n)$ using elementary arguments. Their result can be reinterpreted in terms of Mordell’s theory of representations of binary integral quadratic forms as sums of squares of integral linear forms. Based on this approach, we characterize ${\mathscr S}_m(n)$ for all $m\leq 11$ and provide a few partial results for arbitrary $m$. We also show how Mordell’s results can be used to study variations of the original problem where the sum of the integers is replaced by a linear form in these integers. In this way, we recover and generalize earlier results by Z.-W. Sun et al.

Authors

  • Detlev W. HoffmannFakultät für Mathematik
    Technische Universität Dortmund
    D-44221 Dortmund, Germany
    e-mail

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