How to show that $(2, \sqrt{82})$ in $\mathbb{Z}[\sqrt{82}]$ is not pricipal? I tried the obvious things, like using the norm and trying to show that there were no integer solutions to $a^2 - 82b^2 = 2$, but didn't get anywhere.
(A friend asked me this.)

The equation $$a^2-82b^2=2$$ can be checked for solutions over the integers
by applying continued fractions. See this link :
http://mathworld.wolfram.com/PellEquation.html
