# Numbers greater than Skewes's whose existence can be found in number theoretic proofs

Skewes has proved (without assuming RH) that $\pi(x)<Li(x)$ is violated below $e^{e^{e^{e^{7.705}}}}$ which is clearly a very large number.I was wondering if somewhere else some greater number than Skewes's can be found.

Question :Is there any published proof in number theory breaking this "large number" record?

• I'm guessing you don't consider the upper bounds for the van der Waerden numbers as "number theory". The expression for Skewes' number reminded me of the current known upper bounds for those. – Burak Mar 9 '15 at 17:35
• This seems close to the "eventual counterexamples" question: mathoverflow.net/questions/15444/… – Tobias Fritz Mar 9 '15 at 18:33
• I do not get why the downvote.There is some similar question in MO but not the same.I ask for a new "record". – Konstantinos Gaitanas Mar 9 '15 at 20:28

The quantitative version of the Green--Tao theorem says there is an arithemtic progression of length $k$ in the prime numbers below $$2^{2^{2^{2^{2^{2^{2^{100k}}}}}}}$$ [These are seven 2s, or in non-rendered form for readability 2^2^2^2^2^2^2^(100k)]

• This is close in spirit to the van der Waerden numbers mentioned in a comment by Burak but perhaps a bit more number theoretic. – user9072 Mar 9 '15 at 20:12
• I guess the size of this bound comes from the ineffective nature of Szemerédi's lemma? – David Roberts Mar 10 '15 at 2:02
• Another way of writing it would be $(2\uparrow \uparrow 7)^{100k}$ – fhyve Mar 10 '15 at 2:17
• fhyve, the number you've written looks to be (2^2^2^2^2^2^2)^(100k), which is rather smaller in general. – James Cranch Mar 10 '15 at 8:28
• @DavidRoberts if you mean Szemerédi regularity lemma that would yield still something worse. The argument there is rather along the lines of Gowers's argument for Szemerédi's theorem that avoids this type of argument and thus gets better bounds. At least I think it is like this, I might be wrong though. – user9072 Mar 11 '15 at 0:59

I know that Grahams Numebr used to be biggest one http://en.wikipedia.org/wiki/Graham's_number

Even power towers of the form \scriptstyle a ^{ b ^{ c ^{ \cdot ^{ \cdot ^{ \cdot}}}}} are insufficient for this purpose

-

Graham's number is much larger than many other large numbers such as a googol, googolplex, Skewes' number and Moser's number.

Similiar question can be found at math https://math.stackexchange.com/questions/2497/what-is-the-biggest-number-ever-used-in-a-mathematical-proof

• In case it’s not clear why this answer is getting down-voted: the question specifically asks for large numbers coming from work in number theory. Graham’s Number arose from work in combinatorics, not number theory. – Peter LeFanu Lumsdaine Mar 10 '15 at 1:56