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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

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    $\begingroup$ 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. $\endgroup$ – Gerry Myerson Jun 24 '16 at 8:57
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    $\begingroup$ @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! $\endgroup$ – almagest Jun 24 '16 at 10:02
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See https://oeis.org/A091317 That deals with the case of $n$ prime (and in the text the case $n=p^2$.

Thanks to @GerryMyerson for this (and the reference to AO14662). As he remarks, there is unlikely to be any useful characterisation known. So this is essentially an open problem. But the table does give you several more examples.

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