Nonexistence of Solutions to Certain Families of Diophantine Equations

<p>In this work, I examine specific families of Diophantine equations and prove that they have no solutions in positive integers. The proofs use a combination of classical elementary arguments and powerful tools such as Diophantine approximations, Lehmer numbers, the modular approach, and earlier results proved using linear forms in logarithms. In particular, I prove the following three theorems.</p> <p>Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation</p> <p>(a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2</p> <p>has no solutions in integers X, Y , Z > 1 with a^2X^k ̸= b^2Y^k.</p> <p>Main Theorem II. Let L, M, N ∈ Z+ with N > 1. Then the equation</p> <p>NX^2 + 2^L3^M = Y^N</p> <p>has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1.</p> <p>Main Theorem III. Let p be an odd rational prime and let N, α, β, γ ∈ Z with N > 1, α ≥ 1, and β, γ ≥ 0. Then the equation</p> <p>X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5</p> <p>has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Open Access

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2015

Author dc:creator

• Goedhart, Eva G.

Repository record dc:identifier

https://repository.brynmawr.edu/dissertations/123

OAI identifier oai:identifier

oai:repository.brynmawr.edu:dissertations-1121