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?

Answer 1

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

The number of digits:

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

Answer 2

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.

Answer 3

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

Answer 4

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.

Answer 5

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 $10^{0.302}\times 10^{19728}\sim 2\times 10^{19728}$.

Answer 6

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)