If we think of primes of the form $a^n-b^n,$ where $a,b,n$ are positive natural numbers and $a>b$,

$(a-b)\mid (a^n-b^n)$, so $a-b$ must be $1$

and $n$ must be prime else $(a^r-b^r)\mid (a^n-b^n)$ where $r>1$ and $r\mid n$ 

So, $a^n-b^n$ reduces to $(b+1)^p-b^p$ where $p$ is prime.

If $b=1,(b+1)^p-b^p=2^p-1$ which is well-known Mersenne number.

My questions are :

(1) Why $2$ was chosen for the Mersenne numbers? Is it for the ease of calculation or something else?

(2) Is there any known development on the generalized version i.e., $(b+1)^p-b^p$