On the equation Xⁿ-1=B.Zⁿ

2014/12/18 by Boris Bartolome, Bartolome, Boris, Preda Mihailescu +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1412.5798

arxiv created 2015/06/30 · arxiv updated 2015/07/01

Abstract

We consider the Diophantine equation Xⁿ - 1 = B.Zⁿ, 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: (Xⁿ-1)/(X-1) = n^e.Yⁿ; 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.

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