Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive
integers X and Y. Since there are infinitely many pairs (a, b) for which two
such solutions exist, this result is sharp. It is also effectively proved that
for fixed positive integers a and b, there are at most two positive integer
solutions to the quartic Diophantine equation
aX⁴ - bY² = 2.
We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:BVAU.2429/1288 |
| Date | 11 1900 |
| Creators | Akhtari, Shabnam |
| Publisher | University of British Columbia |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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