Exercise $12$ in Section $1.6$ of Nathanson's : Methods in Number Theory book has the following question.
- When is the sum of a geometric progression equal to a power? Equivalently, what are the solutions of the exponential diophantine equation $1+x+x^{2}+ \cdots +x^{m} = y^{n} \qquad \cdots \ (1)$ in integers $x,m,n,y$ greater than $2$? Check that \begin{align*} 1 + 3 + 3^{2} + 3^{3} + 3^{4} & = 11^{2}, \\\ 1 + 7 + 7^{2} + 7^{3} &= 20^{2}, \\\ 1 + 18 +18^{2} &= 7^{3}. \end{align*} These are the only known solutions of $(1)$.
The Wikipedia link doesn't reveal much about the above question. My question here would be to ask the following:
- Are there any other known solutions to the above equation. Can we conjecture that this equation can have only finitely many solutions?
Added: Alright. I had posted this question on Mathoverflow some time after I had posed here. This user by name Gjergji Zaimi had actually given me a link which tells more about this particular question. Here is the link: