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I'm trying to determine the ideal class group of $\mathbb{Q}(\sqrt{223})$ using elementary methods.

Is there an easy way to show that $a^2 - 223b^2 = - 3$ has no integer solutions? I've tried reducing mod just about everything I could think of, but that doesn't seem to help. Any suggestions?

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Go here and put in the equation and click on "step-by-step" and you will get a proof that there are no solutions. Basically, the left side is so small that any solution has to come from the continued fraction for $\sqrt{223}$, but this has a very short period, and $-3$ doesn't show up.

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