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

Question

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?

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 — JMoravitz, Mar 16 '17 at 17:22
Possible duplicate of What is the order when doing $x^{y^z}$ and why? — Simply Beautiful Art, Mar 16 '17 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. — paw88789, Mar 17 '17 at 02:22
Hmm, The only possible (and simple) way I could think of is Knuth's up-arrow notation — Matthew Roh, Mar 17 '17 at 07:00

score 53 · Answer 1 · answered Mar 16 '17 at 17:17

53

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

The number of digits:

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

answered Mar 16 '17 at 17:17

Jan Eerland

28,671

8

This is all true enough, but it doesn't really answer the question as actually asked. – hBy2Py Mar 16 '17 at 18:59
17

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. – Fluidized Pigeon Reactor Mar 16 '17 at 21:30
The title is a reference for searching and tracking. The question is the text beneath that. NAA. – Nij Mar 17 '17 at 08:43

score 26 · Answer 2 · answered Mar 16 '17 at 17:35

26

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.

answered Mar 16 '17 at 17:35

Daniel W. Farlow

22,531

4

It seems that the Knuth's notation $2\uparrow\uparrow n$ has gained popularity over Rucker's one $^n2$ nowadays. – zwim Mar 16 '17 at 17:47
@Daniel W. Farlow look like you beat me to posting. Got to love it when that happens. – Sentinel135 Mar 16 '17 at 17:53
3

@zwim I actually prefer Rucker's notation, but I do see the appeal of Knuth's very unambiguous notation. – Daniel W. Farlow Mar 16 '17 at 17:55

score 20 · Answer 3 · answered Mar 16 '17 at 19:43

20

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.

answered Mar 16 '17 at 19:43

hmakholm left over Monica

286,031

3

"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.) – Matt Samuel Mar 16 '17 at 22:44
2

@MattSamuel: Milky Way? This number exceeds the number of Planck volumes in the observable universe ... to the hundredth power! – hmakholm left over Monica Mar 17 '17 at 01:24
5

Do you really expect me to do coordinate transformations in my head while strapped to a centrifuge??? – Matt Samuel Mar 17 '17 at 01:25
1

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. – Nayuki Mar 17 '17 at 03:28
1

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. – Martin Rattigan Apr 02 '17 at 21:55
@Nayuki thats still 2^61000 or more magitudes larger...... Which is very large... – Yunfei Chen Feb 02 '21 at 19:43

score 12 · Answer 4 · answered Mar 16 '17 at 23:09

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.
(%i1) 2^2^2^2^2;
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(%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 $10^{0.302}\times 10^{19728}\sim 2\times 10^{19728}$.

There is a mistake in the computed result. The 2 in the middle should be a 3. Probably a typo. — augustin, Mar 17 '17 at 03:43
@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). — celtschk, Mar 17 '17 at 07:59
@celtschk: You, sir, are nitpicking! ;) I didn't mean the one right in the middle, I meant the one after the 7. — augustin, Mar 17 '17 at 09:38

score 6 · Answer 5 · answered Mar 16 '17 at 17:50

6

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 $\lim_{k\to \infty}\;^{(k)}(\sqrt{2})$.

answered Mar 16 '17 at 17:50

Sentinel135

621

Wouldn't that work for any $\sqrt[n]{n}$? – Random832 Mar 16 '17 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 – Sentinel135 Mar 16 '17 at 20:13
Looks like that limit goes to infinity really quickly. – Joshua Mar 16 '17 at 20:36
@Joshua are you sure? can you try and prove it? – Sentinel135 Mar 16 '17 at 20:56
Darn. I hate accumulated roundoff. – Joshua Mar 16 '17 at 21:24

score 4 · Answer 6 · answered Mar 17 '17 at 07:33

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*}

Finally, the answer would be:

\begin{equation*} 2^{65536} \end{equation*}

(correct me if I'm wrong, this was my first answer on Mathematics SE)

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

6 Answers6