If $p^q - 1$ is a prime, then $p=2$ and $q$ is a prime

Question

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem :

If $p$ and $q$ are positive integers ($\mathbb{Z}^+$) such that $q \gt 1$ and $(p^q - 1)$ is a prime, then $p$ is $2$ and $q$ is a prime.

My Question :

I am unable to make any concrete progress . Even a decent hint would be acceptable so that I can build on that ...

Something looks wrong here. $2^i$ is prime if and only if $i=1$. — Ofir Schnabel, Jan 28 '15 at 14:21
$p^q$ cannot be a prime, since it is divisible by $p^n$ for all $0<n<q$. — barak manos, Jan 28 '15 at 14:24
@pranav I gave you an answer below. I left some things undoene by your request. Tell me if you can do it from there on your own. — Ofir Schnabel, Jan 28 '15 at 14:28
Note that the opposite direction is not necessarily correct (with the smallest example being $2^{11}-1=23\cdot89$). — barak manos, Jan 28 '15 at 14:42

score 3 · Accepted Answer · answered Jan 28 '15 at 14:26

3

Since if $p$ is odd then $p^q-1$ is even and larger then two you must have $p=2$. Now, if $q=ab$ then $p^q-1$ is divided by $p^a-1$. (show it). Form here $q$ must be prime.

answered Jan 28 '15 at 14:26

Ofir Schnabel

3,891

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@pranav: It is (at least most of it), but here it's Mathematics community. – barak manos Jan 28 '15 at 14:38
Hi , @OfirSchanbel , thanks a ton :) , I got it :) . The Math.SE community is awesome :) – pranav Jan 28 '15 at 14:43
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You are welcome. – Ofir Schnabel Jan 28 '15 at 14:46
Aren't you are missing the case $p$ even and larger than $2$? – Winther Jan 28 '15 at 15:50

If $p^q - 1$ is a prime, then $p=2$ and $q$ is a prime

1 Answers1