On Exponential Diophantine Triples of Order 2 and the Associated $C^{\infty }$ Differentiable Manifold

We investigate exponential Diophantine triples of order 2, which are sets of three integers $\left\{x,y,z\right\}$ , with $x,y,z>1$ , satisfying $\left(x^2？1\right)\left(y^2？1\right)+1$ , $\left(x^2？1\right)\left(z^2？1\right)+1$ , $\left(y^2？1\right)\left(z^2？1\right)+1$ are perfect squares. It is shown that integer points $\left(x,y,z\right)$ , with $x,y,z>1$ , of a certain $C^{\infty }$ 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.