Artin's conjecture on primitive roots is, of course, wide open. In fact, as many people on this site are aware, it has yet to be proven that, for even a single $n$ satisfying the relevant hypotheses, $n$ is a primitive root modulo infinitely many primes. I was wondering if the following (much weaker) claim was known: are there infinitely many primes $p$ for which the multiplicative order of $2$ mod $p$ is even?
Maybe this is trivial, but I can't see an obvious reason for this to be true. If Artin's conjecture is true then the result follows trivially. It also follows trivially from the existence of infinitely many Fermat primes. I can't find a result of this kind anywhere, which doesn't mean it isn't known, it may just be too trivial to mention in the literature. Does anyone know if this question has an affirmative answer? I restrict to the case $n=2$ for simplicity, but I would be interested in any $n$.
If $2$ is a quadratic non-residue modulo $p$, then its multiplicative order must be even. This is equivalent to $p\equiv\pm3\pmod 8$, and by Dirichlet's theorem there are infinitely many such primes.