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Thue equations and related topics Akhtari, Shabnam
Abstract
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.
Item Metadata
Title |
Thue equations and related topics
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2008
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Description |
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.
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Extent |
768692 bytes
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Genre | |
Type | |
File Format |
application/pdf
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Language |
eng
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Date Available |
2008-08-07
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0066512
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2008-11
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International