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Ben Leggiero's Blog

What Is the Number of Digits in TREE(3)?

I have a question, but I don’t have an answer. I pose this to the mathematics community.

Defining “hops”

For the sake of this question, I will be talking in terms of decimal (AKA base 10).

One trillion (1,000,000,000,000) is 13 digits long. 13 is 2 digits long. 2 is 1 digit long. It took me 3 hops to get to 1 (1,000,000,000,000 ➡️ 13 ➡️ 2 ➡️ 1). Because of this, I say that 1 trillion “has 3 hops”.

One googol (10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000) also has 3 hops (1 googol ➡️ 100 ➡️ 3 ➡️ 1).

One googolplex (I’m not writing that here) has 4 hops (1 googolplex ➡️ 1 googol ➡️ 100 ➡️ 3 ➡️ 1).

Actual question

How many hops does TREE(3) have? It has been proven that it is impossible to store the actual value of TREE(3) in the known universe, but we do know that it is finite. If it is finite, then it has some number of digits.

Is that number of digits small enough to store in the known universe? If not, what about the number of digits in that number of digits?

Finally, how many hops does TREE(3) have? And will the value of that hop number fit in the known universe?

If you can figure this out, please let me know by contacting me at any of my listed contacts. Cheers!