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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 $N^2$.
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참고문헌 (7)
참고문헌 신청
E. Dubouis / 1911 / Solution of a problem of a J. Tannery / Intermediaire Math 18:55~56
E. Grosswald / 1985 / Representations of integers as sums of squares / Springer-Verlag
B. M. Kim / 1998 / On nonvanishing sum of integral squares of Q(√ 5) / Kangweon-Kyungki Math. J 6:299~302
B. M. Kim / 1998 / On nonvanishing sum of integral squares of Q(√2) and Q(√3) / J . Nat. Sci. Res. Inst. KANU 14:1~5
B. M. Kim / 2012 / Sums of Squares of Integers not less than 2 / The Journal of Natural Science, GWNU 18:31~37
E. Grosswald / 1985 / Representations of integers as sums of squares / Springer-Verlag
B. M. Kim / 1998 / On nonvanishing sum of integral squares of Q(√ 5) / Kangweon-Kyungki Math. J 6:299~302
B. M. Kim / 1998 / On nonvanishing sum of integral squares of Q(√2) and Q(√3) / J . Nat. Sci. Res. Inst. KANU 14:1~5
B. M. Kim / 2012 / Sums of Squares of Integers not less than 2 / The Journal of Natural Science, GWNU 18:31~37
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