This thesis is concerned with the resolution of exponential Diophantine equations over the integers, by combining the modular methodology used in the proof of Fermat's Last Theorem with tools coming from computational number theory, such as Thue and Thue--Mahler equations and the primitive divisor theorem on Lehmer sequences. The first two problems that we consider in this thesis are related to the explicit resolution of Diophantine equations. Firstly, we obtain all integer solutions (x, y, n) to the generalised Lebesgue--Nagell equation C_1x^2 + C_2 = y^n, where n > 2, 1 2, 1
Date of Award | 31 Dec 2024 |
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Original language | English |
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Awarding Institution | - The University of Manchester
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Supervisor | Martin Orr (Supervisor) & Gareth Jones (Supervisor) |
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- linear forms in logarithms
- asymptotic Fermat's Last Theorem
- level lowering
- Lehmer sequences
- modularity
- Galois representation
- Frey-Hellegouarch curve
- Exponential Diophantine equation
Applications of the modular method to Diophantine equations
Cazorla Garcia, P. (Author). 31 Dec 2024
Student thesis: Phd