Which finite simple groups have order N so that N+1 is a proper power?
As an example: the simple group of order $168=13^2-1$.



Orders of Finite Simple Groups
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 $\leq 10^{18}$.


