Which finite simple groups have order N so that N+1 is a proper power? As an example: the simple group of order $168=13^21$.

3$\begingroup$ Very few I imagine. But if $N+1$ is a proper power, then the number it is a power of would seem to have nothing much to do with the simple group itself, so what might you learn from a positive answer to the question? $\endgroup$ – Geoff Robinson Jul 4 '14 at 16:17

3$\begingroup$ @Geoff Of course there are the orders which are Mersenne primes. So if you can prove finiteness of the number of solutions for my question then you get finiteness of Mersenne primes. $\endgroup$ – i. m. soloveichik Jul 4 '14 at 16:57

$\begingroup$ Oh, I was thinking of nonAbelian simple groups, excuse me. $\endgroup$ – Geoff Robinson Jul 4 '14 at 17:25
For abelian simple groups your question is merely a disguised form of "enumerate the Mersenne primes". The smallest examples for nonabelian simple groups are as follows:
${\rm PSL}(2,7) + 1 = 13^2$,
${\rm A}_6 + 1 = 19^2$,
${\rm M}_{11} + 1 = 89^2$,
${\rm PSU}(4,2) + 1 = 161^2$,
${\rm J}_1 + 1 = 419^2$
These are all examples with order $\leq 10^{18}$.

1$\begingroup$ I guess you get a few more examples too if you allow quasisimple groups: for example $1+{\rm SL}(2,5) = 11^{2}.$ There is even some kind of "conceptual" explanation for that example, since ${\rm SL}(2,5)$ acts as a Frobenius complement on a $2$dimensional vector space over ${\rm GF}(11).$ $\endgroup$ – Geoff Robinson Jul 4 '14 at 18:03

1$\begingroup$ I can confirm that your `smallest examples' are the only ones with orders below $10^{10}$. $\endgroup$ – Adam P. Goucher Jul 4 '14 at 18:39

1$\begingroup$ @AdamP.Goucher: My search range was $G \leq 10^{18}$, and the examples listed in my answer are all up to that bound. $\endgroup$ – Stefan Kohl Jul 4 '14 at 20:43

1$\begingroup$ @GeoffRobinson and also $1 + 2.A_7 = 71^2$ (because famously $7!+1 = 71^2$). $\endgroup$ – Noam D. Elkies Jul 4 '14 at 21:48