Cyclotomic Norm Diophantine Equations

von Han Chen
Dissertation
Datum der mündl. Prüfung:2023-09-28
Erschienen:2023-10-25
Betreuer:Prof. Dr. Preda Mihailescu
Gutachter:Prof. Dr. Preda Mihailescu
Gutachter:Prof. Dr. Jörg Brüdern
Zum Verlinken/Zitieren: http://dx.doi.org/10.53846/goediss-10158

 

 

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Zusammenfassung

Englisch

In this thesis, I consider two Diophantine norm equations. For the equation of the Nagell-Ljunggren equation $\frac{x^p-1}{x-1} = p^e y^q$ with distinct odd prime exponents $p, q$, I show that, for $p > 3$, it has no solutions under the condition that $q$ does not divide $h_p^-$, the minus part of the class number of the $p$-th cyclotomic field. For the equation of the generalized Ramanujan-Nagell equation $x^2+C=y^p$ with positive integer $C$ and odd prime exponent $p$, I use the theory of Primitive Divisor Theorem for Lucas sequences to find all the positve integer solutions under some congruence conditions.
Keywords: Nagell-Ljunggren equation; The generalized Ramanujan-Nagell equation; Class field theory
 
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