1 . 定义
,其中
为奇素数.
(1)给出同余方程
的满足
的一组解;
(2)(代数基本定理)设
,且
,求证
在
内至多有
个解;
(3)(
小定理)求证:
;
(4)(原根存在定理)若正整数
满足:
,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80754c6842db876978ab0c306640186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411837b4b3078d05b43cb0439259a362.png)
,则记
,则称
为
在
意义下的阶,求证:必定存在
,有
;
(5)求证,存在
,都存在
中必有一者成立;
(6)说明当
时,
必有一组非零解
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fa82bc7122ca89b418d33b694350f987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
(1)给出同余方程
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b2afe64159eac83151200719a5c815.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a964cf4f45603e28dec030a286786750.png)
(2)(代数基本定理)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc2b731f8e2eaa9ac5e3eff49f586820.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a59917a9390d454668b58a0c22c08b84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae6227f2f164910966f194fb857d5e6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86b75c20efa27f38e93e3ce9e00005ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c02bc0c74292b1e8f395f90935d3174.png)
(3)(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c9362bb4f6acf4bf074d0b7bc7b7aba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e34e11a5e27430447b806863c5fdc76e.png)
(4)(原根存在定理)若正整数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e026ac41c5502a4743b336845bae2d95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba1e84f1d28522a63e61283132d1af48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c80754c6842db876978ab0c306640186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411837b4b3078d05b43cb0439259a362.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/097f1e1905c0034991b00c360c5d28f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58bb4eb36d6c696377fe2e14bbb66858.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e81e59019989b7dc2fb59b037ef6e010.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa75d323fa66f6320034f2c7f45a6c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b657e7e787a785063afc56cb983ca56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8e0f45eacebab3cc301c18df8fa0cd1.png)
(5)求证,存在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66543500cdc7031573611000b1b5f85c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/107ba654d76c8b754beb5d173c06c6a7.png)
(6)说明当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b47dded788f1274064925d5f38384e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95730071c97ce5e4d33780626f52e23c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2bad85768515c66beee8beeb0556520.png)
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2 . 一个大于1的整数m,如果对所有的正整数n,都存在正整数x、y、z,使得
,则称m为上数,否则称为下数.试问:是否存在无数多的上数?是否存在无数多的下数?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10663f240e21ee63e7e163ff92919bb0.png)
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