Fekete is opinion on duplicated applications of the tangent function

A high - college pupil called Erna Fekete made an opinion to me using e-mail 3 years earlier, which I can not address. I've given that shed touch with her. I duplicate her intriguing opinion below, in instance any person can give upgraded details on it

Below is just how she phrased it. Allow $b(0) = 1$ and also $b(n)= \tan( b(n-1) )$ . To put it simply, $b(n)$ is the duplicated application of $\tan(\;)$ to 1: $$\tan(1) = 1.56, \; \tan(\tan(1)) = 74.7, \; \tan^3(1) = -0.9, \; \ldots $$

Let $a(n) = \lfloor b(n) \rfloor$ . Her opinion is:

Every integer at some point shows up in the $a(n)$ series.

This series is not unidentified ; it. is A000319 in Sloane is integer series. Basically hers is an inquiry concerning the orbit of 1 under duplicated $\tan(\;)$ - applications. Her and also my examinations at the time led us to think it was an open trouble.

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2019-05-18 23:40:49
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Answers: 2

This isn't an evidence, yet is also wish for a comment, and also might simply be a restatement of the trouble.

For opposition, allow $k$ be any kind of integer such that $b(n) = k$ never ever holds. This suggests $k \leq a(n) < k+1$ never ever holds.

Given that $a(n)$ can not be in between $k$ and also $k+1$, $\arctan a(n)$ can not be either.

Hence, there is a period in between $-\pi/2$ and also $\pi/2$ that a (n) might not touch. Allow is call it $[c,d)$.

Given that tan is routine, $a(n)$ has to additionally stay clear of $m\pi+[c,d)$.

Given that $\pi$ is illogical, $m\pi+[c,d)$ has to have a boundless variety of integers (rather sure this holds true, yet I can be incorrect).

Consequently, there are a boundless variety of periods (coming close to $\pi/2$) that $a(n)$ have to stay clear of. Better, $a(n)$ has to stay clear of the arctans of these periods, and also the arctans of those periods, etc The duplicated arctan periods come close to 0.

Certainly, $a(n)$ additionally needs to stay clear of those periods plus any kind of numerous of $\pi$.

This non - evidence in fact relates to any kind of period $a(n)$ misses out on, so, if real, reveals that $a(n)$ is thick in $\mathbb{R}$. Hope that aids.

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2019-05-19 19:03:25
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I had actually made the very same opinion as Fekete, evidently around the very same time - - mid - 2007. In 2008 I validated that the first twenty million terms do not include 319. (I in fact pressed the confirmation better, yet I can not locate the extra current documents presently.)

Due to the fact that $\tan(x) - x = x^3/3 + O(x^5)$ , the function invests a great deal of its time in a tiny area around $0$ . It runs away when it nears $\pi/2$ and also promptly returns for several models.

A primarily - inexplicable sensation probably pertaining to the above: there are long periods of handful adhered to by brief, 'effective' extends with lots. $\tan^k(1)$ is "listed below 20 approximately" (according to a 2008 e-mail I sent out) for $360110\le k\le1392490$ yet in the next 2000 numbers there are 5 which are over 20.

Extra concept is required!

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2019-05-19 18:04:19
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