Condition on solutions to Pell's equations

Question

If $X_0$ and $Y_0$ be a solution to the Pell's equation of the form $X^2-DY^2=\pm 1$, are $X_0$ and $Y_0$ co-prime? If so, how to prove it?

Answer 1

Suppose $\gcd(X_0,Y_0)=n>1$. Then $n|X^2,DY^2\implies n|X^2-DY^2=1$, a contradiction. Therefore $\gcd(X_0,Y_0)=1$.