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Skewes' number

In number theory, Skewes' number is the smallest natural number x for which the comparison π(x) < Li(x) fails, where π(x) is the prime counting function and Li(x) is the logarithmic integral. That such a number exists is not at all clear; in fact, all available numerical evidence seems to suggest that π(x) is always less than Li(x). However, J. E. Littlewood proved in 1914 that there is such a number (and so, a first such number); and indeed found that the sign of the difference changes infinitely often. But Littlewood's proof did not exhibit a concrete such number x.

Skewes proved in 1933 that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < Li(x) below (now sometimes called first Skewes' number), which is approximately equal to . In 1955, without assuming the Riemann hypothesis he managed to prove that there must exist a value of x below (sometimes called second Skewes' number).

These enormous upper bounds have since been reduced considerably. Without assuming the Riemann hypothesis, te Riele in 1987 proved an upper bound of 7×10370.

Skewes' task was to make Littlewood's existence proof effective: to exhibiting some concrete 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.