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After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in proving it. I'd like to know if it has an actual name, or whether this is just something rather trivial and pointless.

Given a prime $p$ of the form $2n+1$, for all $a$ in the natural numbers if $p$ does not divide $a$, then $p$ divides $a^n+1$ or $a^n-1$.

*Assume non-trivial $a$

Thanks very much.

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Every odd prime number is of the form $2n+1$, by the way. So it's every prime number except $2$. – Asaf Karagila May 5 '13 at 20:18
Fermat's little theorem. – Ma Ming May 5 '13 at 20:19
@AsafKaragila I was trying to make it clear. Stating it as an odd prime means I have to waste a sentence with defining n. But if there's a more elegant way of stating this, I'm all ears. – Thoth19 May 5 '13 at 20:41
up vote 4 down vote accepted

This is well known. Let $p = 2n + 1$ be an odd prime. Then, it is known that if $\gcd(a,p) = 1$, then $a^{p-1} = a^{2n} \equiv 1 \pmod{p}$, by Fermat's Little Theorem. Hence, $(a^n)^2 \equiv 1 \pmod{p}$ so that $a^n \equiv \pm 1 \pmod{p}$.

You do need to know that $x^2 \equiv 1 \pmod{p}$ has exactly two solutions when $p > 2$ is prime, but this is a consequence of $\mathbb{Z}_p$ (The integers mod $p$) being a (finite) field. When $p = 2$, we have $1 \equiv -1 \pmod{2}$ so that $x^2 \equiv 1 \pmod{p}$ has only one solution in this case.

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Thank you. I haven't used that one often enough or recently enough to recognize it. – Thoth19 May 5 '13 at 20:43

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