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"For any prime

*p*, p! is congruent to p

^{2}-p modulo p

^{2}."

Thanks much.

- Thread starter numbthenoob
- Start date

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"For any prime

Thanks much.

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However, if I understand correctly...Euler's theorem is about the relationship between coprimes. I'm researching numbers that are not coprime, for example 7! and 7

To take another example, can I say with certainty that 66797! is congruent to 66797*66796 mod 66797

I've noticed that the pattern holds at least up to 13, i.e.

2! = 2 mod 4

3! = 6 mod 9

5! = 20 mod 25

7! = 42 mod 49

11! = 110 mod 121

13! = 156 mod 169

My question is: ...and so on? And a follow-up: if so, why?

Staff Emeritus

Science Advisor

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p!=p

So really the question is, why is (p-1)!=p-1 (mod p). If you know that Z

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"[URL [Broken] Theorem[/URL]

Last edited by a moderator:

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Thanks Office Shredder and Gib Z, those were both very enlightening comments.

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Good points Office_Shredder I learned something new formula.

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