#### Degree Date

2015

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Abstract

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.

Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation

(a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2

has no solutions in integers X, Y , Z > 1 with a^2X^k ̸= b^2Y^k.

Main Theorem II. Let L, M, N ∈ Z+ with N > 1. Then the equation

NX^2 + 2^L3^M = Y^N

has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1.

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

X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5

has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1.

#### Citation

Goedhart, Eva G. “Nonexistence of Solutions to Certain Families of Diophantine Equations.” PhD diss., Bryn Mawr College, 2015.

## Comments

For those outside the Bryn Mawr community who want access to this dissertation, check Proquest Digital Dissertations, order through your library's ILL department, or see if the dissertation is available for purchase through Proquest