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.

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