New bounds and conditions for the equation of Nagell-Ljunggren
The equation of Nagell-Ljunggren is \[ {x^p-1\over x-1}=p^e \cdot y^q, \quad x, y \in \mathbb Z^*, \quad e\in \{0,1\}, \] where \(p \not= q\) are odd primes. We do not know whether the set of solutions \((p,q,x,y)\) is finite or not. The author proves very strong bounds and conditions on \(q\), for \(p\) fixed, in particular \(q<(p-1)^2\), and moreover \[ \min \{| x| ,| y| \} < q^{10\,p^2}. \] The proof uses clever methods of cyclotomy, and local and global considerations.
scientific article; zbMATH DE number 5151591