Abstract
This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.
Keywords
Diophantine equation
Integer (computer science)
Conjecture
Mathematics
Divisor (algebraic geometry)
Exponential function
Diophantine set
Prime factor
Legendre's equation
Discrete mathematics
Combinatorics
Pure mathematics
Prime (order theory)
Mathematical analysis
Computer science
Subject Areas
Algebraic Geometry and Number Theory
·Geometry and Topology ·Physical Sciences
Analytic Number Theory Research
·Algebra and Number Theory ·Physical Sciences
Polynomial and algebraic computation
·Computational Theory and Mathematics ·Physical Sciences
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