Show that the equation $\frac{x^n-1}{x-1}=4y^2$ has only two solutions

Question

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!

I fear $\dfrac{x^4-1}{x-1}=4y^2$ has solutions $(1,1)$ and $(7,10)$. — Piquito, May 22 '19 at 02:14
@Piquito : As given, the recipe for the LHS is undefined for $x = 1$. — Eric Towers, May 22 '19 at 02:49
@Eric Towers: always LHS is equal to one polynomial for all of the asked $n$. — Piquito, May 22 '19 at 02:55
@Piquito : The LHS and that polynomial agree for all values of $x$ where both are defined. The LHS is undefined for $x = 1$. Consequently, they are not the same thing; the LHS is not the polynomial you are thinking of. — Eric Towers, May 22 '19 at 02:57
Last comment.- Do you say the proposed problem is not the same of putting $x^{n-1}+x^{n-2}+\cdots+x+1=4y^2$? What about of if in your form putting $x=1$ then all $y$ could be solution?. Regards and good by. — Piquito, May 22 '19 at 03:07
@inequality As far as I'm aware there are no notifications sent to question posters when an answer to their question is edited. Just wanted to inform you that new helpful information has been added to my answer. — quantus14, May 24 '19 at 05:17

quantus14 · Answer 1 · 2019-05-24T02:27:34.967

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

Show that the equation $\frac{x^n-1}{x-1}=4y^2$ has only two solutions

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