Recently I've interested in the question: What is the relation between squares and sum of geometric series? In particular, I'm studying the following diophantine equation:
$a^n -1 = (a - 1) m^2$
I'm looking for positive solutions.
I want to know:
- Are there any bases $a$ with infinite number of solutions?
- Is there any efficient method to find nontrivial solutions for a given base?
I was able to prove unsolvability for specific bases $a$ and powers $p$, but not in general.
For instance, the case of $a = 3$ has two positive solutions: $p = 2, m = 2$ and $p = 5, m = 11$.
Can you help me with the general case? I'm trying to apply the theory of cyclotomic polynomials, but no luck so far.
This is a specific instance of the Nagell-Ljunggren equation, and this specific instance is solved. The solutions $$(a,m,n)=(3,11,5),\qquad\text{ and }\qquad (a,m,n)=(7,20,4)$$ listed in the comment above are the only integral solutions with $a,m,n>1$, see also here: https://www.jstor.org/stable/24493413