Fermat proved that if $N = 2^k+1$ is prime, then $k=2^n$ for some $n \geqslant 0$. In this case $N$ is known as a Fermat prime. Only five known Fermat primes exist, corresponding to $n \in \{0,1,2,3,4\}$, and it is thought that perhaps these are the only ones which exist.

Are there any results known for when $N = 2^k+1$ is a power of a prime? A simple application: are there infinitely many finite fields which have the property that all of their units are $2^k$th roots of unity?

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    $\begingroup$ Please look at Wikipedia, Catalan Conjecture, now a theorem. $\endgroup$ – André Nicolas Mar 4 '15 at 21:00

According to Catalan's conjecture (now a theorem, proven in 2002 by Preda Mihăilescu), the only two non-trivial consecutive perfect powers (numbers $x^a$ and $y^b$ for integers $x,y,a,b > 1$) are 8 and 9. It follows that $N = 2^k + 1$ is a prime power only if $N = 9$ or $N$ is itself a (Fermat) prime.


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