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?
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Suppose $\gcd(X_0,Y_0)=n>1$. Then $n|X^2,DY^2\implies n|X^2-DY^2=1$, a contradiction. Therefore $\gcd(X_0,Y_0)=1$.
auscrypt
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+1; I was going to write something along these lines, but you beat me to it; minor detail: OP said $\pm1$, though I think your proof still works – J. W. Tanner May 23 '19 at 23:09