Finding solutions to equation of the form $1+x+x^{2} + \cdots + x^{m} = y^{n}$
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:
- https://mathoverflow.net/questions/58697/
I liked your question much. The cardinality of the solutions to the above equation purely depends upon the values of $m,n$.
Let me break your problem into some cases. There are three cases possible.
- When $ m = 1 $ and $ n = 1 $ , you know that there are infinitely many solutions .
- When $m=2$ and $n=1$ you know that a conic may have an infinitely many rational points or finitely many rational points. In more broad sense, these are genus -1 curves. Where the elliptic curves are also included ( when $m=2,n=3$ or hyper elliptic curves when $m=2, n\ge 4$ ) . This case the number of points on the curve are figured out using the conjecture of Birch and Swinnerton-dyer. It gives you a measure of Cardinality, whether infinite or finite by considering the $L$-functions associated to the curves.
- When $m \ge 2 , n \ge 4$ it may represent some higher dimensional curve. So by the standard theorem of Falting, it has finitely many points given that the curve has genus $g \ge 2$ .
Thank you. I update this answer once if I find something more interesting.