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 the conjecture (given in the latter qouted paper) is used to solve a problem in group theory concerning the non-cyclic graph of a group. Also see Proposition 7.6 of the folloiwng paper for an slightly special case of the conjecture which its solotion has some applications in group theory.

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

Title	Goormaghtigh conjecture
Canonical name	GoormaghtighConjecture
Date of creation	2013-03-22 15:18:55
Last modified on	2013-03-22 15:18:55
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Conjecture
Classification	msc 11J86
Classification	msc 11J61