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I must prove that with $a = \frac12(1+i\sqrt7)$ we have : $|\mathrm{Re}(a^n)| \to \infty$

I tried to use linear algebra by writing $u_n,v_n = Re(a^n), Im(a^n)$ and solve the homogeneous linear recurrence to find a new expression of $u_n$ but I am back to the starting point... The eigen values are as expected $a$ and $\bar{a}$

I also tried to compute $cos(n \times arg(a))$ but it is the same.

Any idea?

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    $\begingroup$ what have you tried? $\endgroup$ – sai-kartik Sep 25 '20 at 13:16
  • $\begingroup$ Can you compute the magnitude $a$? $\endgroup$ – Mark Viola Sep 25 '20 at 13:17
  • $\begingroup$ Is there any chance that there would be a typo in the question? It is easy to prove $|a^n|\to\infty$. It seems hard to prove $|\text{Re}(a^n)|\to\infty$ and, if true at all (!), it may come up from the number-theoretic considerations (behaviour in the ring $\mathbb Z\left[\frac{1+i\sqrt{7}}{2}\right]$). Where's this problem coming from? $\endgroup$ – Stinking Bishop Sep 25 '20 at 13:31
  • $\begingroup$ A friend asked me the riddle. It is in a subject of a math competition competition so the level can be hard. It's been 2 days that we are ten trying to find the solution $\endgroup$ – badinmaths Sep 25 '20 at 13:36
  • $\begingroup$ see oeis.org/A002249 which led me to find the duplicate question... $\endgroup$ – GEdgar Sep 25 '20 at 13:54