# For which odd positive integer $n$, is it true that $−1$ is not a positive power of $2$ modulo $n$? [request for more extensive results]

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 $$\mathbf{Z}_n$$ ?

This question was asked in For which odd positive integer $n$ , is it true that $-1$ is not a positive power of $2$ modulo $n$?

There, the following sequence was referenced: https://oeis.org/A091317

However, the sequences given only cover $$n$$ prime. What if $$n$$ is not prime?

This is not completely trivial, as $$n=15$$ does not have $$-1$$ as a positive power modulo $$n$$, even though $$n=3$$ and $$n=5$$ do. That is, $$3$$ divides $$2^n + 1$$ for some $$n$$, and $$5$$ divides $$2^n + 1$$ for some $$n$$, but $$15$$ does not divide $$2^n + 1$$ for some $$n$$. At the same time, $$33$$ does divide $$2^n + 1$$ for some $$n$$ (as do $$3$$ and $$11$$, obviously).

Is there a way to apply this sequence to composites and prime powers?

• These are tabulated at oeis.org/A014657 My comment at the earlier question, about the llikelihood of a useful characterization, still holds. – Gerry Myerson Aug 12 at 23:59
• Thank you for an answer to the non-prime case. The original question only had the link to the prime case. – Edwin Karat Aug 14 at 3:23
• "This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question." @Arturo, that's exactly what Edwin did. This question should not have been closed. I have reopened it. – Gerry Myerson Aug 14 at 4:15
• @GerryMyerson: I was not aware my vote would close it by itself. – Arturo Magidin Aug 14 at 5:15
• @Arturo, right, I'll edit the title. The question has the group-theory tag, and you have a group-theory gold badge, so one vote from you is all it takes to close (and one vote from me is all it took to reopen, since it has the number-theory tag, and I have a number-theory gold badge). – Gerry Myerson Aug 14 at 6:49

It seems unlikely to me that anyone can say very much that's useful about those values of $$n$$.