On a question of Witno concerning hypothetical elite primes for Mersenne numbers and repunits (Q6134901)

The repunits base \(b\) are \(M_{b,q}=\frac{b^q-1}{b-1}\), \(q\geq 1\), which coincide with the generalized Mersenne numbers when \(q\) is a prime, and in particular with the ordinary Mersenne numbers \(M_{2,q}\). We are concerned with prime numbers \(p\) such that \(M_{b,q}\) is a quadratic residue mod \(p\) for all sufficiently large primes \(q\). Such are called hypothetical elite primes, in reference to the original elite primes, which are associated with the Fermat numbers, not Mersenne. This article is a response to a challenge question: it is now settled that the reciprocal sum of hypothetical elite primes is finite, having their counting function of size \(O\left(\frac{x}{(\log\log x)^2\log x}\right)\).