7
$\begingroup$

Let $u_n=\displaystyle\Re\left( \frac{1+i\sqrt{7}}{2}\right)^n$

Prove that $|u_n| \to \infty$

This appeared in a recent issue of French Revue de la Filière Mathématiques, as it was reportedly asked during an oral exam.

The editor deems it is an "extremely difficult problem", and uses Skolem–Mahler–Lech theorem to prove the result.

They're also looking for a proof an undergraduate could find.

Do you know one ?

$\endgroup$
4
  • 2
    $\begingroup$ It amounts to showing $$2^{n/2}\cos(n\tan^{-1}(\sqrt{7})) \to\infty $$ ...which does appear very difficult. $\endgroup$ – David P Oct 7 '14 at 4:46
  • 4
    $\begingroup$ It might help to write the recurrence: $$u_{n+1} = u_n - 2u_{n-1},u_0=1, u_1=\frac{1}{2}$$ $\endgroup$ – Thomas Andrews Oct 7 '14 at 4:57
  • $\begingroup$ I tried proving this with SML by myself. Even with SML, it is difficult. $\endgroup$ – Sungjin Kim Oct 7 '14 at 17:05
  • 2
    $\begingroup$ I flag it to duplicate because Noam Elkies's proof is 'elementary' but I wouldn't say that it can be done by an undergraduate.. $\endgroup$ – Krokop Oct 7 '14 at 17:15