REPRESENTATION OF A POSITIVE INTEGER BY A SUM OF LARGE FOUR SQUARES

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 nitely 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 nitely many positive integers which cannot be represented by a sum of k squares at least $N^2$.