## Do we know the value of $3 \uparrow\uparrow\uparrow 3$

I was studying Graham’s number and before we can even start calculating $g_1$ which is: $g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$, I was wondering if we even have the actual value of: $3 \uparrow\uparrow\uparrow 3$. I know it is a power tower of 3’s that is 7.6 trillion high (which would reach from the earth to sun … Read more

## Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?

A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat’s last theorem. The question has since been deleted but I was curious as to whether or not there are integer solutions to the equation. More importantly though, the mention of Fermat’s last theorem got me thinking. … Read more

## How to compute the indefinite integrate of n-th tetration of x?

How can the following indefinite integral be computed ? ∫x↑↑ndx where n = {x ∈ N+ : x>2} Here x↑↑n refers to nth tetration of x. I tried searching over the internet to find a general formulae for calculating it when the value of n varies and came across a single paper which had an … Read more

## Iterated logarithm to the n−1n – 1 of the auto tetration n^^n

I’m considering the sequence 1 log(22) log(log(333)) log(log(log(4444))) I tried solving this using Python and I get lots of log(x<0) problems. I thought they were mostly fence post and recursion type errors, so I debugged my code but still the latest iteration has those problems for n>5 or so. I’m beginning to suspect that the … Read more

## Power tower of infinite height mod n

So this one’s been bothering me for a while and I can’t figure it out Define ^kb as b^{b^{b^{…}}} as the power tower of b of height k What I want to do is understand the behavior of \lim_{k \to \infty} {}^kb mod n, ideally to find a closed form. Obviously for integers b \geq … Read more

## Fixed Point of xn+1=ixnx_{n+1}=i^{x_n} [duplicate]

This question already has answers here: Complex towers: iii… (2 answers) Closed 5 years ago. For x∈C, let f(x)=ix=exp(iπx), where i2=−1. Then find the fixed points for f. EDIT: Let for all n≥1 an=ii⋯i⏟n times My question is, does the sequence of tetrations {an}n≥1 converge to some complex number? If yes, then what is it? Answer … Read more

## How do I write Grahams number

I found that Graham’s number is : So, can we say that it is equal to 3^x with x is a power tower of 63 3’s? Answer Knuth’s up-arrow notation works as follows : a\uparrow b=a^b a\uparrow \uparrow b=a\uparrow a\uparrow …\uparrow a \uparrow a with b a’s a\uparrow \uparrow \uparrow b=a\uparrow \uparrow a \uparrow \uparrow … Read more

## How iterated exponential \exp^{[\circ x]}(y)\exp^{[\circ x]}(y), y\neq 1y\neq 1, defined based on tetration?

Background: The tetration $$^xe = \exp^{[\circ x]}(1) = \underbrace{e^{e^{\cdot^{\cdot^e}}}}_{x \text{ times}}$$ is well defined when x \in \mathbb{Z}. The extension of tetration to real height x \in \mathbb{R} can also be understood (though not unique). For instance ^xe \approx 1+x for -1 < x \leq 0, and this can be iterated to interpret … Read more

## Is \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}} rational, algebraic irrational, or transcendental?

Let x=\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\ldots}}}. Assume x is algebraic irrational. By the Gelfond-Schneider Theorem, x^x, which is also x, is transcendental, a contradiction. But I have no idea how to do the rest. Answer The only reasonable meaning of \sqrt 2^{\sqrt2^{\sqrt2^{\cdots}}} would seem to be the limit of the sequence \sqrt2, \sqrt2^{\sqrt2}, \sqrt2^{\sqrt2^{\sqrt2}}, \ldots which can also be … Read more

## Mathematical function for the powers

I have this formula 222…22⏟ni.e. where the total number of 2’s is n. Is there any way to write it as a single mathematical function? Answer Knuth invented a notation for these kinds of expressions, called “up-arrow notation“. To express the power tower in your question with up-arrow notation, we can simply write 2↑↑n. AttributionSource … Read more