All Questions

Is there some general technique to compute tetration and pentation mod some number? $m \uparrow^2 n \pmod a$ and $m \uparrow^3 n \pmod a$ I know about Euler's theorem to compute $m \uparrow n \pmod a$...

$\newcommand\tetra[2]{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ...

$\DeclareMathOperator\len{len}$Let $a, b \in \mathbb{N} -\{ 0, 1 \}$ and define ${^{b}a}$ to be $a^a$ if $b = 2$ and $a^{\left(^{b-1}a \right)}$ if $b \geq 3$ (e.g., ${^{3}5} = 5^{\left( 5^5 \right)} =...

Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review. Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...

Years ago, I defined the "congruence speed" (radix-$10$) of the integer tetration $^{b}a$ as $V(a,b)$, which is the number of the new(!) rightmost digits that freeze when we move from $b \in ...

It is well-known that (in radix-$10$) Graham's number, $G$, can be expressed as a tetration with base $3$ and a very large hyperexponent $\tilde{b}$. Thus, we can write that $\exists! \hspace{1mm} \...