Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 1996; 33(3): 651-655

Printed September 1, 1996

Copyright © The Korean Mathematical Society.

A lower bound for the number of squares whose sum represents integral quadratic forms

Myung-Hwan Kim and Byeong-Kweon Oh

Seoul National University and Seoul National University

Abstract

As a generalization of the famous four square theorem of Lagrange, Mordell and Ko proved that every \po integral \qu of $n$ variables is \re the sum of $n+3$ squares for $1\le n\le 5$. And then for $n=6$, Ko conjectured that every \po integral \qu of six variables that can be represented by a sum of squares is \re the sum of nine squares. In this article, we prove that the conjecture is not valid. We also give, for every $n$, a lower bound for the number of squares whose sum represents all such forms of $n$ variables.

Keywords: sums of squares, positive definite $\Bbb Z$-lattices, representations

MSC numbers: 11E20, 11E25