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

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.

28 pages, 5 figures, Tijdeman-Zagier Conjecture / Beal's Conjecture / Generalized Fermat's Theorem

References in corpus (3)

• Perfect Powers: Pillai's works and their developments
• Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell }+y^{2m}=z^p$
• The Search for Maximal Values of min(A,B,C) / gcd(A,B,C) for A^x + B^y = C^z