Prime Factorial Conjecture: Investigating p! Mod p2 for Prime Numbers

[numbthenoob]

Is there a name and/or proof for the following conjecture?

"For any prime p, p! is congruent to p²-p modulo p²."

Thanks much.

 

[Dickfore]

I think this might be of interest:

"[URL

 

[numbthenoob]

Thanks, Dickfore, that was of interest, even if most of it was over my head.

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 7², where the gcd is 7, not 1.

To take another example, can I say with certainty that 66797! is congruent to 66797*66796 mod 66797² without having to calculate 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?

 

[Office_Shredder]

It might help to reduce the problem

p!=p²-p (mod p²) means that p² divides (p!-p²+p)=p((p-1)!-p+1). We can divide a p out from everything

So really the question is, why is (p-1)!=p-1 (mod p). If you know that Z_p is a field you can figure this out. If not, you probably still can, I just don't see how off the top of my head

 

[Gib Z]

"[URL Theorem[/URL]

 

[numbthenoob]

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

 

[aprillove20]

Good points Office_Shredder I learned something new formula.