# How do you calculate $2^{2^{2^{2^{2}}}}$?

From information I have gathered online, this should be equivalent to $2^{16}$ but when I punch the numbers into this large number calculator, the number comes out to be over a thousand digits. Is the calculator wrong or is my method wrong?

• The number is equal to $2^r$ where $r=2^{16}$. Mar 16, 2017 at 17:17
• this is quiet not clear use parentesis Mar 16, 2017 at 17:17
• If parenthesis are not used, it is assumed that exponents are evaluated from top-down as opposed to bottom-up. $(a^b)^c=a^{bc}\neq a^{(b~^c)}$. Exponentiation is not associative. The answer of $2^{16}$ is for if it were evaluated bottom-up as (((2^2)^2)^2)^2 instead of top-down which is 2^(2^(2^(2^2))) which is much larger Mar 16, 2017 at 17:22
• Possible duplicate of What is the order when doing $x^{y^z}$ and why? Mar 16, 2017 at 21:11
• My TI calculator has an inline option of showing this as 2^2^2^2^2, which evaluates as 65536 (left to right evaluation). But when I switch to math print mode it shows the tower of powers, which it tries, but fails, to evaluate top down...overflow. I'm not saying that a TI calculator is the final arbiter of math truth, but it is what I get. Mar 17, 2017 at 2:22

$$2^{2^{2^{2^2}}}=2^{2^{2^4}}=2^{2^{16}}=2^{65536}\tag1$$

The number of digits:

$$\mathcal{A}=1+\lfloor\log_{10}\left(2^{65536}\right)\rfloor=19729\tag2$$

• This is all true enough, but it doesn't really answer the question as actually asked. Mar 16, 2017 at 18:59
• It currently does answer the question in the title. Equation 1 shows how to evaluate the expression. It also implies, though doesn't explicitly states that the answer the asker's method (or at least the answer he got using the method) was wrong. Mar 16, 2017 at 21:30
• The title is a reference for searching and tracking. The question is the text beneath that. NAA.
– Nij
Mar 17, 2017 at 8:43

What you have is a power tower or "tetration" (defined as iterated exponentiation). From the latter link, you would most benefit from this brief excerpt on the difference between iterated powers and iterated exponentials.

The comment by JMoravitz really gets to the heart of the matter, namely that exponential towers must be evaluated from top to bottom (or right to left). There actually is a notation for your particular question: ${}^52=2^{2^{2^{2^{2}}}}$. You really need to look at ${}^42$ before you get something meaningful because, unfortunately, $${}^32=2^{2^{2}}=2^4=16=4^2=(2^2)^2;$$ however, $${}^42=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}\neq2^8=(4^2)^2=((2^2)^2)^2.$$ Hence, your method is wrong, but everything in those links should provide more than enough for you to become comfortable with tetration.

• It seems that the Knuth's notation $2\uparrow\uparrow n$ has gained popularity over Rucker's one $^n2$ nowadays.
– zwim
Mar 16, 2017 at 17:47
• @Daniel W. Farlow look like you beat me to posting. Got to love it when that happens. Mar 16, 2017 at 17:53
• @zwim I actually prefer Rucker's notation, but I do see the appeal of Knuth's very unambiguous notation. Mar 16, 2017 at 17:55

By convention, the meaning of things written $\displaystyle a^{b^{c^d}}$ without brackets is $\displaystyle a^{\left(b^{\left(c^d\right)}\right)}$ and not $\left(\left(a^b\right)^c\right)^d$.

This is because $\left(\left(a^b\right)^c\right)^d$ equals $a^{b\cdot c\cdot d}$ anyway, so it makes pragmatic sense to reserve the raw power-tower notation $\displaystyle a^{b^{c^d}}$ for the case that doesn't have an alternative notation without parentheses.

As others have explained, $\displaystyle 2^{2^{2^{2^2}}}$ interpreted with this convention is $2^{65536}$, a horribly huge number, whereas $(((2^2)^2)^2)^2$ is $2^{16}=65536$, as you compute.

• "Horribly huge number" rubs me the wrong way in this context. Huge numbers are horrible when they count something that you don't want. The number's not horrible when it's your bank balance! (Actually it might be. If you put together that many pennies it'd probably collapse into a black hole the size of the Milky Way.) Mar 16, 2017 at 22:44
• @MattSamuel: Milky Way? This number exceeds the number of Planck volumes in the observable universe ... to the hundredth power! Mar 17, 2017 at 1:24
• Do you really expect me to do coordinate transformations in my head while strapped to a centrifuge??? Mar 17, 2017 at 1:25
• 2^65536 really isn't a horribly huge number when you consider that its binary representation fits in a mere 8 KB of memory. I mean, we are already using RSA moduli around 2^4096 already, so this number is only around 10× longer. Mar 17, 2017 at 3:28
• Actually all finite numbers are pretty small because all but a finite number of the rest are bigger. Come to that all transfinite numbers are pretty small as well. Apr 2, 2017 at 21:55

I would calculate it using Maxima (which evaluates repeat exponentiation correctly, right-to-left), since there is no point wasting brain cells on something that a machine can do:

$maxima Maxima branch_5_39_base_2_gc9edaee http://maxima.sourceforge.net using Lisp GNU Common Lisp (GCL) GCL 2.6.12 Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report() provides bug reporting information. 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4717124577965048175856395072895337539755822087777506072339445587895905719156736 (%i2) bfloat(%); (%o2) 2.003529930406846b19728 (%i3)  Of course if I just wanted to estimate the magnitude of the number without resorting to the use of arbitrary precision computer software, I'd note that the exponent is$2^{2^{2^2}}=2^{(2^{(2^2)})}=2^{(2^4)}=2^{16}=65536$; multiplying it by$\log_{10} 2\sim 0.30103$gives$19728.302$, so the result is approximately$10^{0.302}\times 10^{19728}\sim 2\times 10^{19728}$. • Wow. Someone actually posted the digits. Mar 16, 2017 at 23:36 • There is a mistake in the computed result. The 2 in the middle should be a 3. Probably a typo. Mar 17, 2017 at 3:43 • "2 in the middle"... can you be a bit more specific? Mar 17, 2017 at 3:48 • @augustin: The digit in the middle is an 8. And yes, I checked it; with the help of the computer, of course. Given that the total number of digits (19729) is odd, the digit in the middle is well defined (it is the digit which is preceded and followed by the same number of digits). Mar 17, 2017 at 7:59 • @celtschk now you sir are fun at parties! Mar 17, 2017 at 8:44 This looks an awfully close to what is known as a tetration (a.k.a. power tower). This is$^{(k)}a=a^{^{(k-1)}a}$where$^1a=a$. For numbers greater than one, these usually get really big really fast, and faster than exponents do. So in your case, you have$^52=2^{2^{16}}$. Now if you want to see an interesting one look at$\lim_{k\to \infty}\;^{(k)}(\sqrt{2})$. • Wouldn't that work for any$\sqrt[n]{n}$? Mar 16, 2017 at 20:04 • Wouldn't what work for any$\sqrt[n]{n}\$? I never said the answer. And yes, but it depends on what you think the answer is. ;D Mar 16, 2017 at 20:13
• Looks like that limit goes to infinity really quickly. Mar 16, 2017 at 20:36
• @Joshua are you sure? can you try and prove it? Mar 16, 2017 at 20:56
• Darn. I hate accumulated roundoff. Mar 16, 2017 at 21:24

Your equation can be simplified using Knuth's up arrow notation. \begin{equation*} 2^{2^{2^{2^2}}} = 2 \uparrow\uparrow 5 \end{equation*}

(because we can calculate tetration with Knuth's up arrow notation)

By definition of Knuth's up arrow notation, You can get this result.

\begin{equation*} 2\uparrow\uparrow5 = 2^{(2^{(2^{(2^2)})})} \end{equation*}

And according to web2.0calc,

\begin{equation*} 2^{(2^{(2^{2})})} = 65536 \end{equation*}