Isaac Asimov featured the Skewes number in his science fact article "Skewered!"
(1974).
In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper bound
was subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966
(Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987),
and less than
(Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil
2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson
left open the possibility that the inequality could fail around , and thus established a large range of violation around
(Derbyshire 2004,
p. 237). More recent work by Demichel establishes that the first crossover occurs
around ,
where the probability that another crossover occurs before this value is infinitesimal
and can in fact be dramatically reduced in the suspect regions where there is such
a risk and these results are almost certainly the best currently possible (P. Demichel,
pers. comm., Aug. 22, 2005).
Rigorously, Rosser and Schoenfeld (1962) proved that there are no crossovers below , and this lower bound was subsequently
improved to
by Brent (1975) and to
by Kotnik (2008).
In 1914, Littlewood proved that the inequality must, in fact, fail infinitely often.
The second Skewes number
(Skewes 1955) is the number above which must fail assuming that the Riemann
hypothesis is false. It is much larger than the Skewes number ,