Richard Guy, Unsolved Problems in Number Theory, 3rd Ed., section D10, writes,
The conjecture of Goormaghtigh, that the only solutions of$${x^m-1\over x-1}={y^n-1\over y-1}$$with $x,y>1$ and $n>m>2$ are $\{x,y,m,n\}=\{5,2,3,5\}$ and $\{90,2,3,13\}$ is still open; some results have been obtained by Le Mao-Hua....
Guy cites several papers of Le Mao-Hua:
On the Diophantine equation ${x^3-1\over x-1}={y^n-1\over y-1}$, Trans Amer Math Soc, 351 (1999) 1063-1074, MR 99e:11033 [but see Leu Ming-Guang and Li Guan-Wei, The Diophantine equation 2x^2+1=3^n$, Proc Amer Math Soc, 131 (2003) 3643-3645 (electronic)].
The exceptional solutions of Goormaghtigh's equation ${x^3-1\over x-1}={y^n-1\over y-1}$, J Jishou Univ Nat Sci Ed, 22 (2001) 29-32, MR 2002d:11036.
On Goormaghtigh's equation ${x^3-1\over x-1}={y^n-1\over y-1}$, Acta Math Sinica, 45 (2001) 505-508, MR 2003f:11045.
Exceptional solutions of the exponential Diophantine equation ${x^3-1\over x-1}={y^n-1\over y-1}$, J Reine Angew Math, 543 (2002) 187-192, MR 2002k:11042.
Actually, Guy cites over 40 papers of Li Mao-Hua, some of which have titles like "An exponential Diophantine equation" and might possibly be concerned with the equation we are discussing.
The Wikipedia page on Goormaghtigh's conjecture, https://en.wikipedia.org/wiki/Goormaghtigh_conjecture contains some more references.