Quick question! if I have $\varphi= (1 +\sqrt{5})/2$ and $\psi = (1 - \sqrt{5})/2$ and $n\ge0$ would be an int
How would I prove that in $(\varphi^n - \psi^n)/\sqrt{5}$ is an integer but only using Newtons binomial theorem?
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2$\begingroup$ Do you realize that $\;\psi\phi=-1\;$ ? $\endgroup$– DonAntonioFeb 4, 2014 at 20:21
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$\begingroup$ well I have to use binomal therom and the values are placed with the equation . $\endgroup$– user125035Feb 4, 2014 at 20:28
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1$\begingroup$ Expand $\varphi^n$ with the binomial theorem, just keeping a $2^n$ as the denominator. Now do the same thing for $\psi^n$. Note that the terms are the same up to sign, when you subtract all the ones not involving $\sqrt{5}$ cancel each other out. See where you can go from there. $\endgroup$– NateFeb 4, 2014 at 21:15
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