Diophantine approximation and the equation (a^2 c x^k - 1)(b^2 c y^k - 1) = (abc z^k - 1)^2

TL;DR

This paper proves that a specific exponential Diophantine equation has no solutions in positive integers under certain conditions, advancing understanding in number theory related to exponential equations.

Contribution

It establishes the non-existence of solutions for a class of exponential Diophantine equations with constraints on variables and exponents.

Findings

No solutions for x, y, z > 1 when k ≥ 7 and a^2x^k ≠ b^2y^k.

The proof applies number-theoretic methods to exclude solutions.

Results contribute to the theory of exponential Diophantine equations.

Abstract

We prove that the Diophantine equation (a^2 c x^k - 1)(b^2 c y^k - 1) = (abc z^k - 1)^2 has no solutions in positive integers with x, y, z > 1, k \geq 7 and a^2x^k \neq b^2y^k.