On the equation X^n-1=B.Z^n

TL;DR

This paper investigates solutions to the Diophantine equation X^n - 1 = B.Z^n, linking solutions to properties of Euler's totient and the Nagell-Ljunggren equation, and identifies families with no solutions.

Contribution

It establishes conditions under which solutions exist or do not exist, connecting the equation to the Nagell-Ljunggren equation and recent results, and provides parametrized families with no solutions.

Findings

Solutions imply divisibility conditions involving Euler's totient.

Solutions relate to the diagonal case of Nagell-Ljunggren equation.

Identifies parametrized families with no solutions.

Abstract

We consider the Diophantine equation X^n - 1 = B.Z^n, where B in Z is understood as a parameter. We prove that if the equation has a solution, then either the Euler totient of the radical, phi(rad (B)), has a common divisor with the exponent n, or the exponent is a prime and the solution stems from a solution to the diagonal case of the Nagell-Ljunggren equation: (X^n-1)/(X-1) = n^e.Y^n; e = 0 or 1. This allows us to apply recent results on this equation to the binary Thue equation in question. In particular, we can then display parametrized families for which the Thue equation has no solution. The first such family was proved by Bennett in his seminal paper on binary Thue equations.