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

Abstract

In this paper, we consider the equation(1)By assuming the abc conjecture is true, in [8], Luca and Walsh gave a theorem, which implies that the equation (1) has only finitely many solutions n, x if a and b are different fixed positive integers. We solve (1) when m = 1 and (a, b) ∈ {(2, 10), (4, 100), (10, 58), (3, 45)} . Moreover, we show that (a n − 2)(b n − 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 (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 n − 2)(b n − 2) = x 2 has a solution n, x, then n ≤ 6, where 2 < a < b.