Abstract

We prove that if (x,y,n,q)≠(18,7,3,3) is a solution of the Diophantine equation (xⁿ − 1) ∕ (x − 1) = y^q with q prime, then there exists a prime number p such that p divides x and q divides p − 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy computer calculations.

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