Korean J. Math. Vol. 24 No. 1 (2016) pp.71-79
DOI: https://doi.org/10.11568/kjm.2016.24.1.71

Representation of a positive integer by a sum of large four squares

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Published: Mar 30, 2016
Keywords:
large square, representation, sum of squares, four square theorem
Mathematics Subject Classification:
11E25, 11E10, 11Z05.

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Byeong Moon Kim
Department of Mathematics Gangneung-Wonju National University Gangneung 210-702, Korea

Abstract

In this paper, we determine all positive integers which cannot be represented by a sum of four squares at least $9$, and prove that for each $N$, there are finitely many positive integers which cannot be represented by a sum of four squares at least $N^2$ except $2\cdot4^m$, $6\cdot4^m$ and $14\cdot4^m$ for $m\ge0$. As a consequence, we prove that for each $k\ge5$ there are finitely many positive integers which cannot be represented by a sum of $k$ squares at least $N^2$.


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