On the Exponential Diophantine Equation $(a^2-2)(b^2-2)=x^2$

TL;DR

This paper investigates a specific exponential Diophantine equation, providing solutions for particular cases, proving non-existence under certain conditions, and proposing conjectures about the nature of its solutions.

Contribution

It offers explicit solutions for special cases, proves non-existence under certain divisibility conditions, and introduces conjectures relating solutions to Pell and Pell-Lucas numbers.

Findings

Solved the equation for specific (a,b) pairs when m=1.

Proved no solutions exist if 2 divides n and gcd(a,b)=1.

Proposed conjectures linking solutions to Pell and Pell-Lucas numbers.

Abstract

In this paper, we consider the equation $(a^n-2^{m})(b^n-2^{m})=x^2$. By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the above equation has only finitely many solutions $n,x$ if a and b are different fixed positive integers. We solve the above equation when $m=1$ and $(a,b)=(2,10),(4,100),(10,58),(3,45)$. Moreover, we show that $(a^2-2)(b^2-2)=x^2$ has no solution n,x if 2|n and gcd$(a,b)=1$. We also give a conjecture which says that the equation $(2^2-2)((2P_k)^n-2)=x^2$ has only the solution $(n,x)=(2,Q_k)$, where $k>3$ is odd and $P_k,Q_k$ are Pell and Pell Lucas numbers, respectively. We also conjecture that if the equation $(a^2-2)(b^2-2)=x^2$ has a solution $n,x$, then $n<7$, where $2<a<b$.