[Thanks to Gerhard Paseman for helping me reformulate my original question.]


The equation
$$
\frac{a^m-1}{a-1}=b^2
$$
was solved by Ljunggren, building on work of Nagell, who showed that if $a>1$, $b>1$, and $m>2$ are integers, there are exactly two (easy to find) solutions. Can the same methods explicitly give all solutions to the following equation?
$$
\frac{a^m-1}{a-1}=2b^2
$$
Also, is this worked out anywhere in the literature? (If not, I'm thinking of giving it as a problem to an undergraduate researcher. But if there is some easy way to approach the problem I'd like to know.)



A modification of the Ljunggren-Nagell equation
The equation 
$$
\frac{a^m-1}{a-1}=2b^2
$$
does not have solutions in positive integers for $m>2$ as shown below.


First, notice that $a$ must be odd.


Second, one can see that $m$ must be even. Indeed, if $a\equiv 3\pmod4$, then odd $m$ would produce an odd number in the left-hand side. On the other hand, if $a\equiv 1\pmod4$, then from Theorem 3 of the [LTE](http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS8wLzgyODNhOGNhOWQ4OWM1NDk5NTY1MGQyNWVlYWNlMzE1OGYxMDM0&rn=TGlmdGluZyBUaGUgRXhwb25lbnQgLSBWZXJzaW9uIDYucGRm) it follows that 
$$
\nu\_2(m)=\nu\_2(\frac{a^m-1}{a-1})=\nu\_2(2b^2)>0,
$$
i.e. $m$ is even.


Let $m=2k$ where $k>1$. Then
$$
\frac{a^m-1}{a-1}=\frac{a^k-1}{a-1}(a^k+1).
$$
Since $\gcd(a^k-1,a^k+1)=2$, we have two cases to consider: $\frac{a^k-1}{a-1}=c^2$ or $a^k+1=c^2$ for some $c\mid b$.


In the former case:


* if $k=2$, we get $(a,c)=(t^2-1,t)$ for any $t$ satisfying $(t^2-1)^2+1=2d^2$ where $d=b/c$. However, it can be computationally verified (e.g., in Magma) that this elliptic curve has the only solution with $t=0$ (giving $a=-1$).
* if $k>2$, we get Ljunggren equation giving $(k,a,c)=(5,3,11)$ and $(k,a,c)=(4,7,20)$, which however do not produce any solutions to the original equation.


In the latter case, one can refer to [Mihailescu's Theorem](https://en.wikipedia.org/wiki/Catalan%27s_conjecture) to obtain the only solution $(k,a,c)=(3,2,3)$, which again does not produce any solutions to the original equation.


Hence, the equation in question does not have any solutions for $m>2$.


