--- In

primenumbers@yahoogroups.com, "mikeoakes2" <mikeoakes2@...> wrote:

>

> Lemma: For positive integer n, the integer

> f(n) = (6*n+2)^(6*n+2) - (6*n+1)^(6*n+1) is never square-free.

>

> Proof: Let w = (-1+sqrt(-3))/2. Then 1+w = -w^2 and

> 3*(12*n^2+6*n+1) = (6*n+2+w)*(6*n+1-w).

> Let P be a rational prime divisor of 12*n^2+6*n+1 and let

> p be a corresponding prime divisor of 6*n+2+w in Q(sqrt(-3)), with

> norm(p) = P. Then 6*n+2 = -w+m*p, for some m in Q(sqrt(-3)), and

> f(n) = (-w+m*p)^(6*n+2) - (w^2+m*p)^(6*n+1).

> Now we expand in p, using the binomial theorem, to obtain

> (-w+m*p)^(6*n+2) = (-w)^(6*n+2)*(1+m*p) mod p^2 and

> (w^2+m*p)*(6*n+1) = (w^2)^(6*n+1)*(1+m*p) mod p^2.

> Since (-w)^(6*n+2) = (-w)^2 and (w^2)^(6*n+1) = w^2, we obtain

> f(n) = 0 mod p^2. Hence f(n) is divisible by norm(p^2) = P^2

> and is hence not square-free.

I believe the Lemma, but not my proof, because:-

Take e.g. n=8;

then the repeated factor of n^n-(n-1)^(n-1) is 19;

but 19 does not divide (12*n^2+6*n+1).

I can't see the error in my algebra [am short of sleep].

Please help.

Mike