# For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$ i.e. $[-1] \ne [2^k] , \forall k >0$ in $\mathbb Z_n$ ? Is there any ( at least sufficient ) characterization for such odd $n$ ?

( I have only found $n=7$ is such a number ) . Please help . Thanks in advance

• The primes for which $-1$ is a power of 2 are tabulated at oeis.org/A014662 with some references listed but I doubt there's any useful characterization known. – Gerry Myerson Jun 24 '16 at 8:57
• @GerryMyerson I have put that as CW. Hope you are content. I am trying to do my bit to cut down on unanswered questions which are answered in comments! – almagest Jun 24 '16 at 10:02

See https://oeis.org/A091317 That deals with the case of $n$ prime (and in the text the case $n=p^2$.