Abstract
This study proves that the Diophantine equation $\left(9d^2+1\right)^x+\left(16d^2-1\right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy's Primitive Divisor Theorem, specifically when $d$ is not divisible by $5$. In cases where $d$ is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.
Keywords
Diophantine equation
Mathematics
Conjecture
Integer (computer science)
Logarithm
Divisor (algebraic geometry)
Left and right
Combinatorics
Pure mathematics
Discrete mathematics
Mathematical analysis
Computer science
Subject Areas
Mathematical Dynamics and Fractals
·Mathematical Physics ·Physical Sciences
Algebraic Geometry and Number Theory
·Geometry and Topology ·Physical Sciences
advanced mathematical theories
·Mathematical Physics ·Physical Sciences
Citations by Year
2024: 1
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