# What is the largest number used in a useful mathematical proof that isn't just an upper or lower bound? [closed]

There are quite a few famous gigantic numbers used in useful mathematical proofs, like Skewes's number, Graham's number, and the number $2 \uparrow \uparrow 10^{10^6}$ from Coward and Lackenby's 2011 result on determining the ambient isotropy of links. However, these numbers are all upper (or lower) bounds, which I think is not quite as interesting, because you can often make an upper bound higher by just doing less work.

What is the largest number to have appeared naturally in a useful mathematical proof that isn't just an upper or lower bound? Obviously the terms "naturally" and "useful" make this question somewhat subjective. Also, I'm not particularly interested in numbers defined in the form "the smallest number satisfying $X$ property", so I'd prefer a number where we can actually estimate its value to some degree of precision - e.g. we can roughly estimate how many iterated logarithms (or another slowly-growing function) would be required to bring it down to $o(1)$. (I think this requirement would rule out $\mathrm{TREE}(3)$.)

One possibility that occurs to me is $\sim 8 \times 10^{53}$, the cardinality of the monster group, but I bet there are larger ones.

## closed as too broad by Peter, Lord Shark the Unknown, max_zorn, onurcanbektas, NamasteAug 17 '18 at 12:29

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Possible duplicate of What is the biggest number ever used in a mathematical proof? – Simply Beautiful Art Aug 10 '18 at 16:23
• I don't believe the linked dupe has any restriction that requires answers to be upper or lower bounds, so your question still falls as a subset of the linked one. I do see that perhaps the result may be different, with much smaller numbers here than the other, due to the restriction, but I still think anything here also belongs there; hence a dupe. – Simply Beautiful Art Aug 10 '18 at 16:36
• – Dave L. Renfro Aug 10 '18 at 17:48
• @catsteevens johncarlosbaez.wordpress.com/2018/03/09/… – tparker Aug 11 '18 at 4:54
• @tparker "useful proof" must definitely be specified. – Peter Aug 11 '18 at 13:03