Why π^π^π^π could be an integer (for all we know!).
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as the standard, accessible explanation of why the integrality of pi^pi^pi^pi is an open problem, and cite it when asking follow-up questions about towers of exponents and precision.
Recommended by 2 people across 2 different months, 2021 to 2025.
What it is
Matt Parker of Stand-up Maths explains why nobody can currently rule out that the tower pi^pi^pi^pi is an integer, even though it is almost certainly not. He covers the size of the number, how many digits of pi would be needed to evaluate it, and transcendental number theory such as Schanuel's Conjecture.
What people said
Recommends Abijah · 2021-02 · Mathematics Stack Exchange · on “Given a real number what is the minimum number of times it can be tetrated to get an intiger?”
I really enjoyed this video [this video] on the possibility that $\pi^{\pi^{\pi^\pi}}$ is an integer, but i thought that it was a case of a more interesting general problem. Given a real $x$ and an integer $y$ , what is the minimum non-zero integer $y$ such that the tower $x↑↑y$ is an integer; where $↑↑$ represents tetration [LINK] (i.e. repeated exponentiation)? Its clear that for integers $y=1$ ; and that for the $n$ th root of a multiple of $n, y$ is at most $n+1$ . For what other real numbers can we obtain $y$ ? Should we expect $y$ to exist for all real numbers?
Recommends Blue · 2025-06 · Mathematics Stack Exchange · on “Why is it so difficult to determine whether or not $\pi^{\pi^{\pi^\pi}}$ is an integer?”
FYI: This is the topic of the 2021 video "Why $\pi$^$\pi$^$\pi$^$\pi$ could be an integer (for all we know!)." [this video] on Matt Parker's "Stand-up Maths" YouTube channel. The upshot: We won't know until we either get better at math (understanding nuances of transcendental numbers such as $\pi$) or else get better at computing (calculating values with sufficiently-great precision).
Cites Mare · 2021-02 · Mathematics Stack Exchange · on “Powers of transcendental numbers that lead to integers”
For a given real number $x>0$ define $a_k(x)$ as follows for $k \geq 1$ : $a_1(x)=x$ , $a_{k+1}(x)=x^{a_k(x)}$ , so that $a_2(x)=x^x$ , $a_3(x)=x^{x^x}$ .... Question 1: Is there a explicit transcendental number $x$ such that $a_k(x)$ is an integer for a $k>0$ ? Question 2: Let $Y:= \{x \in \mathbb{R} \mid x>0$ , $x$ is transcendental and there exists $k >0$ such that $a_k(x)$ is an integer $\}$ . Is $Y$ measurable and if yes, what is its measure? The question is motivated by [this video] where it is discussed whether $a_4(\pi)$ is an integer.
Cites OneCold Ruben · 2021-03 · Mathematics Stack Exchange · on “What methods are used to look into $\pi^{\pi^{\pi^\pi}}$ and other stacked towers of repeating irrational numbers?”
I recently saw a YouTube video here [this video] and it got me thinking, does anyone on this site have any insight on how we would tackle (change to base ten, evaluate, compute, or even put limits on approaching) some things like this? examples: $\displaystyle e^{e^{e^e}}$ $\displaystyle \pi^{\pi^{\pi^\pi}}$ $\displaystyle \sqrt{2}^{\sqrt{2}^{\sqrt{2}^\sqrt{2}}}$ $\ln(2)^{\ln(2)^{\ln(2)^{\ln(2)}}}$ What methods are proposed to even look at these things? Would it be a good idea to change to complex for pi? Do all towers of root 2 just simplify to 2? note: Apologies for being new and not knowing how to format the text correctly. Edit: Things I am thinking, for $$\displaystyle \sqrt{2}^{\sqrt{2}^{\s
Cites u003f · 2021-03 · Mathematics Stack Exchange · on “How many digits of $\pi$ are needed to approximate $\pi^n$ correctly to the nearest whole number?”
In his piece on $\pi^{\pi^{\pi^\pi}}$ , Matt Parker plots a graph [this video] showing that if $\pi$ is approximated to $n$ digits, then whenever $m \leq 2 n$ (roughly), $\pi^m$ is correct to the nearest whole number. Is there any theoretical underpinning for this?
Cites badjohn · 2024-03 · Mathematics Stack Exchange · on “Smallest prime factor of $\lfloor e\uparrow e\uparrow e\uparrow e\rfloor$?”
Matt Parker on YouTube has an article on whether $\pi \uparrow \pi \uparrow \pi \uparrow \pi$ is an integer. Spoiler alert: we don't know, it's too big. [this video]
Cites Semiclassical · 2024-06 · Mathematics Stack Exchange · on “Why is ${(e^{(i×1)})}^π$ different from $e^{(i×π)}$”
As an example of the difficulties posed by exponentiating by $\pi$, there's the fact that we numerically checking that pi^pi^pi^pi is not an integer is beyond our computational reach (as covered by Stand-Up Maths here [this video] ).
Cites Alma Arjuna · 2024-12 · Mathematics Stack Exchange · on “How many digits of $\pi$ must we know in order to determine the value of $\pi^{\pi^{\pi^\pi}}$ up to the first decimal place?”
It is an open problem to decide whether $\pi^{\pi^{\pi^\pi}}$ is an integer or not (though it is obviously not), as mentioned in this video [this video] by Matt Parker. The integer part of this number have almost $10^{18}$ digits (WolframAlpha estimates it as $10^{10^{17.8236}}$ ). For comparison, we only know $10^{14}$ digits of $\pi$ as of this day. If this doesn't seem completely out of reach to modern computing (as we 'only' require $10$ thousand times more computational power), then there is the problem of precision. This raises the following question: How many digits of $\pi$ must we know in order to determine the value of $\pi^{\pi^{\pi^\pi}}$ up to the first decimal place? A comment below