Goormaghtigh conjecture

For integers $x,y,m$ and $n$ satisfying $x>1$, $y>1$, and $n>m>2$, the equation

$$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}$$

has only the solutions $(x,y,m,n)=(5,2,3,5)$ and $(90,2,3,13)$.

See the following paper for some progress’ on the conjecture:

M. Le, Exceptional solutions of the exponential Diophantine equation$(x^3-1)/(x-1)=(y^n-1)/(y-1)$. J. Reine Angew. Math. 543 (2002) 187-192.

See Section 7 of following paper in which the solution of a certain case of theconjecture (given in thelatter qouted paper) is used to solve aproblem in group theory concerning the non-cyclic graph of a group. Also seeProposition 7.6 ofthe folloiwng paper for an slightly special case of the conjecture which itssolotion has someapplications in group theory.

Alireza Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group,Communications in Algebra, 35 (2007) 2057-2081.