# Thread: [SOLVED] Prime number proof

1. ## [SOLVED] Prime number proof

I wasn't sure where to put this, but here goes.

$\displaystyle \frac{n^x - n}{x}$ is a formula we have been given.
We have been told that the denominator is a factor when x is prime and we're expected to prove/show that to be the truth.

It works experimentally (substituting numbers) but I can't actually find a proper proof.

Best I could come up with is:

Let it equal $\displaystyle \frac{a}{b}$ so that therefore b is a factor of a (a=by, y is an integer)

$\displaystyle n^x - n = a = by$
and
$\displaystyle x = b$

That leads me to subsitute them back and get:

$\displaystyle \frac{by}{b}$

which simplifies to give y, which is an integer.

Any proper proofs?

2. Originally Posted by Russ
I wasn't sure where to put this, but here goes.

$\displaystyle \frac{n^x - n}{x}$ is a formula we have been given.
We have been told that the denominator is a factor when x is prime and we're expected to prove/show that to be the truth.

It works experimentally (substituting numbers) but I can't actually find a proper proof.

Best I could come up with is:

Let it equal $\displaystyle \frac{a}{b}$ so that therefore b is a factor of a (a=by, y is an integer)

$\displaystyle n^x - n = a = by$
and
$\displaystyle x = b$

That leads me to subsitute them back and get:

$\displaystyle \frac{by}{b}$

which simplifies to give y, which is an integer.

Any proper proofs?
The numerator factors as follows

$\displaystyle n(n^{x-1}-1)$

If x is prime then by fermat little theorem$\displaystyle x | n^{x-1}-1$

Fermat's little theorem - Wikipedia, the free encyclopedia

3. Thanks. Didn't know about that theorem, was trying to do it with logic.