The motivation for this question comes from the following problem from an international Team selection test of 2007 from Chile:
Problem: Let $p$ be a prime number. Find all pairs of positive integers $m,n$ that satisfy the following equation $$(p-1)(p^n-1)=4m(m+1)$$
I have tried a lot to solve this problem and proposed it in other webpages and I have never seen a solution. Hence, I think maybe the jury who made the exam put an intractable/open problem without realize it.
Notice that the equation in the problem has a solution iff the discriminant (as a quadratic equation in $m$) is a perfect square. This led to study the equation $$(p-1)p^n-(p-2)=x^2.\tag{$*$}$$
So my first question is
Question 1: Given $a,b,c\in \mathbb{Z}$. Is there any systematic study of the Diophantine equation $$ab^n+c=x^2$$ maybe for some particular choice of $a,b,c$?.
By looking mod $p$ equation ($*$) and using quadratic reciprocity one can show that $p$ should be $\pm 1 \text{ mod } 8$ but I am not sure how much one can say.
So let's take a particular example of a difficult prime for this equation, say $p=23$:
Question 2: Does the equation $$22\cdot 23^n-21=x^2$$ has an integral solution for $n>1$?
If someone solve the original problem would great, but an heuristic that it may be open (based in an answer of question 1 for example) would leave me satisfied.
As this may be approached by comparing two ideal factorizations in the ring of integers of $\mathbb{Q(\sqrt{-c})}$ I will put the tag for algebraic number theory.