## Re: [PrimeNumbers] Fermat twin prime theorem

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• Well, I ll take a guess... perhaps (n-1)!+1 is not congruent to 0 mod (n+2)? Joseph. ... __________________________________ Do you Yahoo!? Read only the mail
Message 1 of 2 , Dec 11, 2004
Well, I'll take a guess... perhaps (n-1)!+1 is not
congruent to 0 mod (n+2)?

Joseph.

--- "John W. Nicholson" <reddwarf2956@...>
wrote:

>
>
> == means congruent
>
> Wilson's theorem for twins is
>
> Let n>=2. The integers n and n+2 form a pair of twin
> primes iff
>
> 4[(n-1)!+1] + n == (mod n(n+2))
>
> But,
>
> (n-1)!+1 == 0 (mod n)
>
> and by a Corollary of Fermat treorem with (a,n)=1,
>
> a^(n) - a == 0 (mod n)
>
> so,
>
> a^(n)-a == (n-1)!+1 == 0 (mod n).
>
> Therefore,
>
> 4(a^(n)-a) + n == 0 (mod n(n+2))
> or
> 4*a^(n) - 4*a + n == 0 (mod n(n+2))
>
>
> If it has not been stated before, I am surprised.
>
>
>
>

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