Goormaghtigh Conjecture

Goormaghtigh, Rene. L'Intermediaire des Mathematiciens 24 (1917), 88

The conjecture has been subject to extensive computer supported solution search, especially in small cases

In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh about the solutions of the exponential Diophantine equation x m − 1 x − 1 = y n − 1 y − 1 {\displaystyle {\frac {x^{m}-1}{x-1}}={\frac {y^{n}-1}{y-1}}} with distinct integers x , y {\displaystyle x,y} larger than one and exponents larger than two.