Negative polynomial Pell's equation.

The negative polynomial Pell's equation is $P^2(X)-(X^2+d)Q^2(X)= -1$, where $d$ is an integer. In this paper, we prove that it has no non-trivial integer polynomial solutions $P(X)$ and $Q(X)$ if and only if $ d \not= \pm 1, \pm2$. For $d = \pm1, \pm2$, we investigate the existence of polynomial solutions $P(X), \, Q(X)$ with integer coefficients.