**MSC:**- 11D57

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 .