On the Diophantine equation (x^m-1)/(x-1) = (yⁿ-1)/(y-1)

Yann Bugeaud and T.N. Shorey

Vol. 207 (2002), No. 1, 61–75

DOI: 10.2140/pjm.2002.207.61

Abstract

We study the Diophantine equation x^m−1 x−1 = yⁿ−1 y−1 in integers x > 1, y > 1, m > 1, n > 1 with x≠y. We show that, for given x and y, this equation has at most two solutions. Further, we prove that it has finitely many solutions (x,y,m,n) with m > 2 and n > 2 such that gcd(m − 1,n − 1) > 1 and (m − 1)∕(n − 1) is bounded.

Milestones

Received: 22 January 2001

Published: 1 November 2002