Polynomial Pell’s equations

Abstract:

The polynomial Pell’s equation is ${P^2} - ({x^2} + d){Q^2} = 1$, where $d$ is an integer and the solutions $P,Q$ must be polynomials with integer coefficients. It is proved that this equation has nonconstant solutions if and only if $d = \pm 1, \pm 2$, and in these cases all solutions are determined.