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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 $N^2$.

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