Do we know the value of $3 \uparrow\uparrow\uparrow 3$

Question

I was studying Graham's number and before we can even start calculating $g_1$ which is:

$g_1 = 3\uparrow\uparrow\uparrow\uparrow 3$,

I was wondering if we even have the actual value of:

$3 \uparrow\uparrow\uparrow 3$.

I know it is a power tower of 3's that is 7.6 trillion high (which would reach from the earth to sun of powers of 3's if using a slightly larger font). I know that even the powers of three go up pretty quickly. So, do we even have a ball park figure for this number?

Answer 1

The number of times you have to take the logarithm to make the number handy is about $3^{27}$, smaller than the number Ross Millikan mentions, but still very large.

To be exact, we have

$$10\uparrow\uparrow (3^{27}-1)<3\uparrow\uparrow\uparrow 3=3\uparrow\uparrow (3^{27})<10\uparrow\uparrow (3^{27})$$

If we take $log_{10}$ $3^{27}-2$ times, we arrive at a number between $10$ and $10^{10}$, definitely handy. If we define $10^{10^{10}}$ to be handy, we have to take the logarithm only $3^{27}-3$ times.

This is definitely too much to have any hope to calculate the number. Even the calculation of, lets say, $3\uparrow\uparrow 10$ , is infeasible.

Neither is there any way to comprehend the size of this number. But the size of the power tower can be comprehended.

Answer 2

You have to define what you mean by a ball park figure. If you mean the power of $10$ that represents it, no. Even that power takes far too many digits to write down. Each time you take a log, you take one layer off the stack, so you have to take the log trillions of times to get an understandable number. On the other hand, your description implies an algorithm that could compute it in finite time. Don't hold your breath.

In this question about Graham's number the poster defines a handy number as one less than our age. Why it wasn't a number you can count on your hands, I don't know, but it doesn't change much. In my answer I consider the number you are asking about. A way to get a ball park number for some large numbers is to count the number of times it takes applying the log function to make the number handy. In your case it is $7625597484985$, which is not a handy number. In the comments the $\log^*$ function is defined, which is the number of times you need to apply $\log$ to a number to make it handy, so $\log^* 3 \uparrow\uparrow\uparrow 3=7625597484985$, and the $\log$ of this is about $60$. We can then say $\log \log^* 3 \uparrow\uparrow\uparrow 3$ is handy. This gives a way to compare large numbers.