Graham’s Number Is Too Big for Me to Tell You How Big It Is

I was going to write an April Fool’s Day post with the title “Mathematicians Declare Graham’s Number Equal to Infinity.” Graham’s number is really big, but of course, it’s precisely 0% as big as infinity. On the other hand, everything we touch is finite, so in some sense, Graham’s number is probably “close enough” to infinity for doing most math. I didn’t end up writing that post because Graham’s number is too big for the post to be funny.

For example, here’s one of the paragraphs I would have written:

“The decision to round Graham’s number up to infinity promises to simplify the proof of the twin prime and Goldbach conjectures as well as other longstanding number theory problems: now they can be proved by checking the finite number of remaining cases.”

But this is where I got stuck. I wanted to tell you how many cases might be left to check for one of these number theory problems. Goldbach’s conjecture, which states that every even integer is the sum of two primes, is known to be true for numbers up to 10¹⁸. But Graham’s number minus 10¹⁸ is basically equal to Graham’s number. For all practical purposes, we’re 0% of the way to checking that the Goldbach conjecture is true for numbers up to Graham’s number.

Here is where I should tell you how big Graham’s number is. But I can’t.

Read the full post at Roots of Unity.