Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime.

QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?

In 2014 I conjectured further that for any prime $p>3$ there is a prime $q<p/2$ such that $M_q=2^q-1$ is a primitive root modulo $p$ (cf. http://oeis.org/A236966 ). For example, $M_{17}=131071$ is a primitive root modulo the prime $37$. Of course, the question is easier than this difficult conjecture.

I'm even unable to prove that each prime $p>3$ has a quadratic nonresidue of the form $2^n-1$ with $n$ a positive integer. Any ideas towards the proof?

By the way, in 2014 I also conjectured that for each prime $p>3$ there is a Fibonacci number smaller than $p/2$ which is a quadratic nonresidue modulo $p$ (cf. http://oeis.org/A241568). This should be quite challenging.

Any comments are welcome!