Daniel Grace <[EMAIL PROTECTED]> asks
> I looked at Chris Caldwells page on Wieferich (1909)
> primes but I could not see exaclty how p^2|2^(p-1)-1 relates to
> Mersennes with square factors? I can see that Mp=3(2^(p-1)-1).
> So my question is this "How does one derive Wieferich's result,
> from the statement: let p be a prime and n be an integer such
> that p^2|2^n-1?"
> I assume that n must be a prime otherwise:
> Is it always true that if q|2^p-1 where p & q are primes
> then q^2|2^(pq)-1? eg. 23^2|2^(23.11)-1.
Write 2^p = 1 + k*q, since you assume q | (2^p - 1).
You want to show that q^2 divides (1 + k*q)^q - 1.
Use the binomial theorem. Or show by induction on m that
(1 + k*q)^m == 1 + k*q*m (mod q^2)
for all integers m >= 0. Then set m = q.
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