Is there a proof which shows that for all primes p, n^p-n is divisible by p?

👨🏻‍🏫 學術台數學

[匿名]

2017-11-09 2:29 am

I've shown that for all positive integers n that n^5-5 is divisible by 5.

I've also found that this seems to be the case for p|(n^p-n), given that p is one of the first five primes. And that it is not true that p divides n^p-n when p is a composite positive integer.

更新1:

Excuse me, I've shown that for all positive integers n that n^5-n is divisible by 5. Typo above.

回答 (2)

用戶65385

2017-11-09 5:41 am

✔ 最佳答案

Factor n^p - n into n(n^[ p-1 ] - 1).

Then, according to Fermat's "little" theorem,

p | n^(p-1) - 1

👍 0 👎 0

DWRead

2017-11-09 4:35 am

n^5-5 is not divisible by 5 for n = 1.

👍 0 👎 0

收錄日期: 2021-04-24 00:48:18
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20171108182947AAzRStJ

檢視 Wayback Machine 備份