STNB 2016 (30è any)

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


Boris Bartolomé


We consider the Diophantine equation $X^n-1=B.Z^n$, where $B$ in $\mathbb{Z}$
is understood as a parameter. We prove that if this equation has a
solution, then either the Euler totient of the radical, \varphi (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 in {0;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. (This is a joint work with P.Mihailescu).


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