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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Goedhart, Eva G."],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Open Access","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-01T08:00:00Z","date_published":"2015-01-01T08:00:00Z","updated_at":"2026-07-24T01:23:46Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.brynmawr.edu/dissertations/123","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Goedhart, Eva G."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2015-05-15T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://repository.brynmawr.edu/dissertations/123"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Nonexistence of Solutions to Certain Families of Diophantine Equations"]}]}],"canonical_facts":{"dc:creator":["Goedhart, Eva G."],"dc:date.available":["2015-05-15T07:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://repository.brynmawr.edu/dissertations/123"],"dc:title":["Nonexistence of Solutions to Certain Families of Diophantine Equations"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Open Access"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:23:46Z"}