Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality

TL;DR

This paper proves the Tijdeman-Zagier conjecture by analyzing the irrationality of slopes in a geometric interpretation, demonstrating that no solutions exist for the exponential Diophantine equation with coprime bases.

Contribution

It introduces a novel geometric approach linking slope irrationality and coprimality to prove the conjecture by contradiction.

Findings

No integer solutions for the equation with coprime bases and exponents > 2.

Properties of slopes relate to coprimality and irrationality.

Explicit proof of the conjecture via geometric and number-theoretic arguments.

Abstract

The Tijdeman-Zagier conjecture states no integer solution exists for $A^X+B^Y=C^Z$ with positive integer bases and integer exponents greater than 2 unless gcd$(A,B,C)>1$. Any set of values that satisfy the conjecture correspond to a lattice point on a Cartesian graph which subtends a line in multi-dimensional space with the origin. Properties of the slopes of these lines in each plane are established as a function of coprimality of terms, such as irrationality, which enable us to explicitly prove the conjecture by contradiction.