# 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 GoormaghtighConjecture 2013-03-22 15:18:55 2013-03-22 15:18:55 mathcam (2727) mathcam (2727) 8 mathcam (2727) Conjecture msc 11J86 msc 11J61