Let $\ b\in\mathbb Z,\ $ and $\ |b|>1.\ $ Call $$ M_b(n)\ :=\ \frac{b^n-1}{b-1} $$ to be $n$-th Mersenne number mod $b$. The necessary condition for $\ M_b(n)\ $ to be a prime is that $\ n\ $ is a prime.
The two cases of $\ |b|=2\ $ seem the most attractive.
Let $\ A_b := M_{-b}\ $ for every $\ b>1. $ Call $\ A_b(n)\ $ to be the base $b$ alternative Mersenne numbers.
For b=2 there is the Lucas-Lehmer test.
Question: Is there or can you provide (makeup) a similar test for $\ A_2(p)\ $ alternative Mersenne numbers?
This would very likely produce new large primes. Naively, there seems to be more base 2 alternative Mersenne primes than the classical Mersenne primes. An actual theorem along these lines would be wonderful.
