Preda Mihailescu
Class Number Conditions for the Diagonal Case of the Equation of Nagell -- Ljunggren
Preprint series: Mathematica Gottingensis
The diagonal case of the Nagell-Ljunggren equation is
\[ \frac{x^p-1}{x-1} = p^e \cdot y^p, \quad \hbox{with} \quad x, y \in \Z,
\quad e \in \{0, 1\}, \] and $p$ an odd prime. This is considered to
be the hardest case of equations of the type
\[ \frac{x^p-1}{x-1} = p^e \cdot y^q, \quad \hbox{with} \quad x, y \in \Z,
\quad e \in \{0, 1\}, \] where $p$ and $q$ are odd primes. In this
paper we attack the problem with methods used in the cyclotomic
approach to Fermat's Last Theorem and some additional methods specific
for this case. As a result we derive some general class number
conditions, which are the first known general algebraic necessary
conditions for the equation to have solutions. It follows that the
equation has no solutions for $p < 12 \ 000 \ 000$; the previous known
upper bound was $p \leq 17$.

We also give explicite upper bounds for the solution $|x|$, which are
necessary, independently of the class number conditions .
Keywords: Diagonal Nagell - Ljunggren Equation