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Twisted extensions of Fermat's Last Theorem Bruni, Carmen Anthony
Abstract
Let x, y, z, n, α ∈ ℤ with α ≥ 1, p and n ≥ 5 primes. In 2011, Michael Bennett, Florian Luca and Jamie Mulholland showed that the equation involving a twisted sum of cubes [equation omitted] has no pairwise coprime nonzero integer solutions p ≥ 5,n ≥ p²p and p ∉ S where S is the set of primes q for which there exists an elliptic curve of conductor NE ∈ {18q,36q,72q} with at least one nontrivial rational 2-torsion point. In this dissertation, I present a solution that extends the result to include a subset of the primes in S; those q ∈ S for which all curves with conductor NE ∈ {18q,36q,72q} with nontrivial rational 2-torsion have discriminants not of the form ℓ² or -3m² with ℓ,m ∈ ℤ. Using a similar approach, I will classify certain integer solutions to the equation of a twisted sum of fifth powers [equation omitted] which in part generalizes work done from Billerey and Dieulefait in 2009. I will also discuss limitations of the methods for these equations and as they extend to further prime exponents.
Item Metadata
Title |
Twisted extensions of Fermat's Last Theorem
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2015
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Description |
Let x, y, z, n, α ∈ ℤ with α ≥ 1, p and n ≥ 5 primes. In 2011, Michael Bennett, Florian Luca and Jamie Mulholland showed that the equation involving a twisted sum of cubes [equation omitted] has no pairwise coprime nonzero integer solutions p ≥ 5,n ≥ p²p and p ∉ S where S is the set of primes q for which there exists an elliptic curve of conductor NE ∈ {18q,36q,72q} with at least one nontrivial rational 2-torsion point. In this dissertation, I present a solution that extends the result to include a subset of the primes in S; those q ∈ S for which all curves with conductor NE ∈ {18q,36q,72q} with nontrivial rational 2-torsion have discriminants not of the form ℓ² or -3m² with ℓ,m ∈ ℤ. Using a similar approach, I will classify certain integer solutions to the equation of a twisted sum of fifth powers [equation omitted] which in part generalizes work done from Billerey and Dieulefait in 2009. I will also discuss limitations of the methods for these equations and as they extend
to further prime exponents.
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Genre | |
Type | |
Language |
eng
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Date Available |
2015-04-21
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivs 2.5 Canada
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DOI |
10.14288/1.0166239
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2015-05
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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-NoDerivs 2.5 Canada