"Very Large Numbers"

Question

A couple days ago I was bored and thought of the following problem and am hoping someone can either answer my question or point me in the right direction.

Is there an easy way to compute the number of digits of the following? Follow up is, can the number of digits be presented in an easily presentable/human readable way?

googolplex knuth's up-arrow googolplex

References:

• https://en.wikipedia.org/wiki/Googolplex
• https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

Answer 1

One uparrow in Knuth's notation is just regular exponentiation, so you are asking about $N=(10^{10^{100}})^{10^{10^{100}}}=10^{(10^{100}\cdot {10^{10^{100}}})}$ which has $10^{100}\cdot {10^{10^{100}}}+1$ digits.

This is actually not such a large number. If we take $\log_{10} (\log_{10} (\log_{10} N)))$ we get about $100$. In this answer I consider $3 \uparrow \uparrow \uparrow 3$ and find it takes $7625597484985$ applications of the $\log$ to make the number handy. We define $\log^*$ as the number of applications of $\log$ to make a number handy, so we have $\log^* N=4$, but $\log^*3 \uparrow \uparrow \uparrow 3=7625597484985$ so it takes $\log \log^*3 \uparrow \uparrow \uparrow 3$ to get handy. Bigger numbers will need $\log^{**}$, the number of times you need to apply $\log^*$ and so on.

Answer 2

At the risk of being snarky, there is a simple way to write out the number of digits:

$$ 1 + \lfloor \log_{10}\left( \text{googolplex} \uparrow \uparrow \text{googolplex}\right) \rfloor $$

Answer 3

If you just mean a single up arrow, then this is the same as computing a googol plex to the power of a googol plex. If I let $g$ denote the ordinary googol ($=10^{100}$), and $G$ denote the googol plex ($=10^{10^{100}}$), then the number of digits is the log base $10$ of $G^G$ plus $1$ (since both $g$ and $G$ are powers of $10$): \begin{align} \mathrm{Num\ Digits} &= 1 + \log_{10} \left(G^G \right) \\ &= 1 + G \log_{10} G \\ &= 1 + G \log_{10} \left[10^\left(10^{100} \right) \right] \\ &= 1 + G \cdot 10^{100} \\ &= 1 + Gg \end{align} So the number of digits is a googol plex times a googol, plus $1$.

Or to put it another way: the number of digits is one plus $1$ followed by a googol zeros and another hundred zeros.