# Limit of $|\mathrm{Re}(a^n)|$ where $a=\frac12(1+i\sqrt7)$ [duplicate]

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?

• what have you tried? – sai-kartik Sep 25 '20 at 13:16
• Can you compute the magnitude $a$? – Mark Viola Sep 25 '20 at 13:17
• 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? – Stinking Bishop Sep 25 '20 at 13:31
• 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 – badinmaths Sep 25 '20 at 13:36
• see oeis.org/A002249 which led me to find the duplicate question... – GEdgar Sep 25 '20 at 13:54