1 . 设
为一个质数,且
也是一个质数,证明:
的小数表示形式中包含0至9的所有数码.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ca4c3cdbb63bf2e3c1c3d67d46d3acc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cbadc8f735cb509c99741c41f32d097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a1eccdf91bcfb339dee1e546ebcf17bc.png)
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2 . 对每个正整数n,定义
为从1到n中所有与n不互质的正整数的和.求证:若
且
,则
是合数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38fcec7af3520884b173b29bda6c657a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a9ba50ede8ef97b843accf839fef5c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7d9712c3b25f3030e166e136d3a4686.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae9dd56551121a5b88be4dc6910c9ec0.png)
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3 . 正整数
,且
的素因子个数不超过2,对于任意整数
,若
,则有
成立,求证:
是质数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0704f453b2de48d36911f7db496bbf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2794218743c2cccfe26d67c803ca094b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d7c6588252b11da61d1bb7f9b7766a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
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