Goormaghtigh conjecture

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


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 equationMathworldPlanetmath (x3-1)/(x-1)=(yn-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 PropositionPlanetmathPlanetmathPlanetmath 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