Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren

  • Preda Mihăilescu
Part of the Developments in Mathematics book series (DEVM, volume 16)


The diagonal case of the Nagell-Ljunggren equation is
$$ \frac{{x^p - 1}} {{x - 1}} = p^e \cdot y^p with x,y \in \mathbb{Z} e \in \left\{ {0,1} \right\}, $$
and p an odd prime. The only known nontrivial solution is
$$ \frac{{18^3 - 1}} {{18 - 1}} = 7^3 , $$
and it is conjectured to be also the only such solution. However, it is not even proved that (1) has only finitely many solution.


Exponential diophantine equation diagonal case of Nagell and Ljunggren 

2000 Mathematics subject classification

11D61 11D45 


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© Springer-Verlag 2008

Authors and Affiliations

  • Preda Mihăilescu
    • 1
  1. 1.Mathematisches InstitutUniversität GöttingenGöttingenGermany

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