The Diophantine equation $b^2X^4-dY^2=1$

Abstract

If $b$ and $d$ are given positive integers with $b > 1$, then we show that the equation of the title possesses at most one solution in positive integers $X,Y$. Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.