I have an equation of the form
$\displaystyle 2(2x-1)^2 -(2n-1)^2 = 1$
I know that this is a pell's equation, but I don't know what to do with it, since its not in the form.
$\displaystyle x^2 - dn^2 = 1$
Thanks for your help!
Let $\displaystyle a=2n-1 $ and $\displaystyle b=2x-1 $. We then have $\displaystyle a^2-2b^2=-1 $
Thus $\displaystyle (a^2-2b^2)^2=(-1)^2=1 \implies (a^2-2b^2)^2-2(2ab)^2=1 $.
But $\displaystyle 2b^2=a^2+1 $, so we get $\displaystyle (2a^2+1)^2-2(2ab)^2=1 $.
Summarizing, we see our original equation was transformed into Pell's equation.