Nonexistence of Solutions to Certain Families of Diophantine Equations

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 \u3e 1 with a^2X^k ̸= b^2Y^k. Main Theorem II. Let L, M, N ∈ Z+ with N \u3e 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 \u3e 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