On Exponential Diophantine Triples of Order 2 and the Associated C ∞ Differentiable Manifold

Abstract

We investigate exponential Diophantine triples of order 2, which are sets of three integers { x , y , z } , with x , y , z > 1 , satisfying ( x 2 − 1 ) ( y 2 − 1 ) + 1 , ( x 2 − 1 ) ( z 2 − 1 ) + 1 , ( y 2 − 1 ) ( z 2 − 1 ) + 1 are perfect squares. It is shown that integer points ( x , y , z ) , with x , y , z > 1 , of a certain C ∞ differentiable manifold form such triples. This paper establishes a recursive method for generating, via successive mutations (operations analogous to Vieta mutations as in the Markov surface), infinite families of these triples, thereby linking number theory with differential geometry.