TITLE:
On Exponential Diophantine Triples of Order 2 and the Associated C ∞ Differentiable Manifold
AUTHORS:
Mouftaou Adjibade, Joachim Moussounda Mouanda
KEYWORDS:
Diophantine Equation, Exponential Diophantine -Tuple of Order, Differentiable Manifold, Normalized Triple, Markov Surface, Vieta Mutation
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.16 No.3,
March
25,
2026
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.