# Fekete’s conjecture on repeated applications of the tangent function

A high-school student named Erna Fekete made a conjecture to me via email three years ago,
which I could not answer. I’ve since lost touch with her.
I repeat her interesting conjecture here, in case anyone can provide updated
information on it.

Here is how she phrased it. Let $$b(0)=1b(0) = 1$$ and $$b(n)=tan(b(n−1))b(n)= \tan( b(n-1) )$$.
In other words, $$b(n)b(n)$$ is the repeated application of $$tan()\tan(\;)$$ to 1:
$$tan(1)=1.56,tan(tan(1))=74.7,tan3(1)=−0.9,…\tan(1) = 1.56, \; \tan(\tan(1)) = 74.7, \; \tan^3(1) = -0.9, \; \ldots$$

Let $$a(n)=⌊b(n)⌋a(n) = \lfloor b(n) \rfloor$$.
Her conjecture is:

Every integer eventually appears in the $$a(n)a(n)$$ sequence.

This sequence is not unknown; it
is A000319 in Sloane’s integer sequences.
Essentially hers is a question about the orbit of 1 under repeated $$tan()\tan(\;)$$-applications.
Her and my investigations at the time led us to believe it was an open problem.

Because $$tan(x)−x=x3/3+O(x5)\tan(x) - x = x^3/3 + O(x^5)$$, the function spends a lot of its time in a small neighborhood around $$00$$. It escapes when it nears $$π/2\pi/2$$ and quickly returns for many iterations.
A mostly-unexplained phenomenon presumably related to the above: there are long spans of small numbers followed by short, ‘productive’ spans with large numbers. $$tank(1)\tan^k(1)$$ is “below 20 or so” (according to a 2008 email I sent) for $$360110≤k≤1392490360110\le k\le1392490$$ but in the next 2000 numbers there are five which are above 20.