If $$(7+4\sqrt{3})^n = p+\beta,$$ where $n$ and $p$ are positive integers and $\beta$ is a proper fraction, then show that $$(1-\beta)(p+\beta)=1.$$
I cant even understand how to express the term in a positive number and a proper fraction. I would appreciate any hint.
Fun question! The key realization is that $$ (7 + 4\sqrt{3})^n + (7 - 4\sqrt{3})^n \in \mathbb{Z} $$ (do you see why?) and moreover, that $$ 0 < (7 - 4\sqrt{3}) < 1, $$ so that $$ 0 < (7 - 4\sqrt{3})^n < 1, $$ for all natural numbers $n$. It follows from here that \begin{align*} p &= (7 + 4\sqrt{3})^n + (7 - 4\sqrt{3})^n - 1 \\ \beta &= 1 - (7 - 4\sqrt{3})^n. \end{align*} Now to finish, we see directly from the above that \begin{align*} p + \beta &= (7 + 4\sqrt{3})^n \\ 1 - \beta &= (7 - 4\sqrt{3})^n. \end{align*} Multiply them together and see what you get.