Skewes' number

Skewes' number, or the first Skewes number, is a large number. Assuming that the Riemann hypothesis is true, it is an upper bound for the first prime number for which the comparison π(x) < Li(x) must fail, where π(x) is the prime counting function[?] and Li(x) is the logarithmic integral. Its value is <math>e^{e^{e^{79}}}</math>, which is approximately equal to <math>10^{10^{10^{34}}}</math>.

Another large number, the second Skewes number, is an upper bound for the same thing, assuming that the Riemann hypothesis is false. Its value is <math>10^{10^{10^{1000}}}</math>, which is a lot bigger than the first Skewes number.

Riele[?] reduced the first upper bound to <math>e^{e^{27\over4}}</math>; in the other direction Conway and Guy[?] claim a lower bound of <math>10^{1167}</math>.

J. E. Littlewood proved that there was such a number (and so, a first such number); and indeed found that the sign of the difference changes infinitely often. There was then the question of making this result effective: of exhibiting some upper bound for the first sign change. According to Kreisel, this was at the time not considered obvious even in principle. The approach called unwinding in proof theory looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Skewes's result was celebrated partly because the proof structure used excluded middle, which is not a priori a constructive argument (it divides into two cases, and it isn't computable in which case one is working).

Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as Graham's number.