Define $x§y$ as the power tower : $x^{x^x...}$ where $...$ means $y$ times. For instance $2§1=2,2§2=4,2§3=16,2§4=2^{16}$.
See : http://en.wikipedia.org/wiki/Tetration
Let $p$ be a prime larger than $100$. Let $n$ be an integer such that $0 < n < p$.
How many residue classes does $2§n$ mod $p$ have ?
Lets define $R(f(n),p$) as the number of residue classes for $f(n)$ mod $p$.
Conjecture : $R(2§n,p)< p - log(p) - slog(p)$
where the $log$ and $slog$ are the base $2$ logarithm and super-logaritm.
See : http://en.wikipedia.org/wiki/Super-logarithm
I got this question from the tetration forum. It was posted by tommy1729. Below is the link:
http://math.eretrandre.org/tetrationforum/showthread.php?tid=833
I assume the conjecture is true and can be much improved. Is this true ?
Same question for the strongly related case : $2à1=2$,$2à n = (2à(n-1))$ mod $p$.