In a previous post I asked for primes $p$ of the form: $$p=\frac{7^{q}+1}{7^{q-n^2}+1},$$ where $q$ and $n$ are positive integers.
The solutions found up to $q=5000$ are $[1,17,4]$, $[8,24,4]$, $[2,38,6]$ and $[4,148,12]$. In brackets the second number is $q$, the first number is $q-n^2$ and the third number is $n$.
Do you believe that the next solution is extremely huge or even far beyond the programs and calculators capacities?
Do you believe that the number of these primes is not infinite?
Do you believe that a prime of this type exists such that $q-n^2\neq 2^k$ where $k$ is an integer $\geqslant 0$?
Here is the link to the previous question