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!

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- Apr 24th 2010, 08:07 PMdavismjPell's equation
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! - Apr 24th 2010, 09:03 PMmr fantastic
- Apr 24th 2010, 09:21 PMdavismj
- Apr 24th 2010, 11:02 PMchiph588@
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. - Apr 24th 2010, 11:05 PMchiph588@
- Apr 25th 2010, 05:06 PMdavismj
- Apr 25th 2010, 05:10 PMchiph588@
- Apr 25th 2010, 05:22 PMdavismj