A new like on my post made me remember the number I had in mind.

I’ll admit now that I was bluffing about the proof, but I have a concrete natural number* in mind and would like to test my hypothesis. I know there’s some mathematicians or otherwise mathematically advantaged people around here who could challenge this statement. Anyone willing to do so?

My only two rules, is that the number can be unambiguously defined from your text alone, and that you’ll have to give a reference to an amateur or professional mathematician having mentioned the number you propose (or a provably larger one, including proof) in some capacity involving a general public, like a publication, or a talk, or whatever. This is to avoid the “your number +1” kind of trick. Feel free to use self-reference.

My number needs 3 symbols to describe (5 if brackets are included), and about a paragraph of rather universally understandable mathematics to define.

*) Although, as a set theorist, I would’ve been happy with giving a staggeringly large infinite quantity, I think the question, in spirit, is meant for finite numbers. If you want to have an infinite challenge, I’m happy to take it as well.

I mean, if it is somehow possible for any individual with access to your post, given an arbitrarily large amount of hypothetical time and space, to deduce what specific natural number you are referring to, then go ahead and post your answer.

Perhaps the rule in spirit is about being able to show that your very large number is larger than the very large numbers of the other people. Thus, making something unwieldy will be in your disadvantage, since it will complicate the “proof” part of the game.

Also, you don’t have to come up with the proof yourself, you’ll win if you mention the largest number. (But only after a proof has been provided, by anyone)
Perhaps we can make it a community effort.

I’m interested to see where this could lead.

And bonus points for succinctness and originality, of course!

If I take pi and I multiple it by each digit coming after the 3 . Because the decimal part is infinite, if you give a number supposed to be the biggest I can multiply my own number to be still bigger

I think the idea is to come up with a concrete number in mind, not an algorithm on how to take someone else’s number and go a bit beyond it with a few steps more of your algorithm.

But the product of the (nonzero) digits of pi tends to infinity. The limit of increasingly large natural numbers is generally considered (or better said, defined) to be the smallest kind of infinite quantity.

If this is your entry in the infinite-number game, it would be the equivalent of giving the answer “1” in the finite number game.

On the other hand, if it is an entry to the finite-number game, then the problem is that it is not a finite number.

It is, because it is much more natural (at least for me) to assume there is no largest finite number.

Suppose X is the largest finite number, then by the obviously true “two finite quantities combined are still finite”, also X+1 is a finite number. But, then X < X+1… So then X was not the largest finite number.

Therefore, if we assume the existence of a largest finite number, we’d have to discard the (to me) rudimentary belief that the combination of two finite quantities will be finite.

However, the task is not to find the largest finite number per se, but to find a larger finite number than the other people in this thread.

Let f_{0}(n)=n+1. Define by induction f_{k+1}(n)=f_{k}^{n}(n) (the power means n-fold composition).
Then f_{9}(9) is already quite big. But the definition is a bit too simple, it must be possible to do better.

It could be left as something to chose, as long as the rules of calculation work. I mean I was thinking of these finite groups on which you define x^(n+1)=1

A bit larger than the 42 that I originally wanted to post. I wanted to beat the already fairly impressive 1001 and I don’t think 42 would have had any chances.