Prove or disprove
Let $x,y,n$ be postive integers, show that the equation given below has only two solutions $$\frac{x^n-1}{x-1}=4y^2$$
I have found that $(x,n,y)=(3,2,1)$ and $(x,n,y)=(7,4,10)$ are solutions to the above equation. However, there maybe other solutions.
Thanks!
There are infinitely many solutions. $(x,n,y)=(4y^2-1,2,y)$ For all $y$. (The op used $y=1$ already.)
Edit: The op's question is asking when is the expression $\frac{x^n-1}{x-1}$ is an even square. The more general question of when the expression is a power number has been heavily investigated. When $n>2$ it is conjectured that there are only three solutions $(x,n)=(18,3),(7,4),(3,5)$. Only one of these results in an even square (which already has been stated by the op). Partial results of the general problem shows that there are no other solutions to the op's problem. More information can be found here: https://mathoverflow.net/questions/58697/a-geometric-series-equalling-a-power-of-an-integer