92年國安局三等考數理組初等數論(四)

Posted by mathfat — 十二月 13, 2010

題目：證明對所有整數 $x, y$ , $\displaystyle \frac{x^2+y^2-3}{4}$ 不是整數。

解答：Suppose we have know that

For every integer $n$ , then $n^2\equiv 1$ or $0$ mod 4.

Then $x^2+y^2-3 \equiv 3, 2$ or $1$ mod 4.

So $\displaystyle \frac{x^2+y^2-3}{4}$ is not an integer

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