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

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

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Published: Mar 30, 2016

Keywords:

large square, representation, sum of squares, four square theorem

Mathematics Subject Classification:

11E25, 11E10, 11Z05.

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 \cdot 4^{m}

,

6 \cdot 4^{m}

and

14 \cdot 4^{m}

for

m \geq 0

. As a consequence, we prove that for each

k \geq 5

there are finitely many positive integers which cannot be represented by a sum of

k

squares at least

N^{2}

.

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