Skewes number

The first Skewes number, written \(Sk_1\), is an upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) is true, where \(\pi(n)\) is the and \(li(n)\) is the . This bound was first proven assuming the .Skewes Number It's equal to \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\).Edward Kasner, James Roy Newman. Mathematics and the Imagination Originally published by Simon and Shuster, 1940. Dover Edition published in 2001. ISBN 978-1556151040 p.32Conway and Guy. The Book of Numbers. Copernicus. 1995. ISBN 978-0387979939 p.145 The number is named after Stanley Skewes, who found the bound in 1933.
The second Skewes number, \(Sk_2\), is a closely related upper bound for the least number \(n\) such that \(\pi(n) > li(n)\) holds, but this bound, as opposed to the previous one, was proven without assumption of the Riemann hypothesis. It is equal to \(e^{e^{e^{e^{7.705}}}}\) ~ \(10^{10^{10^{963}}}\), which is larger than the original Skewes number.
As of now, it is known that the least example \(n\) of \(\pi(n) > li(n)\) must lie between \(10^{19}\) and \(1.4 \cdot 10^{316}\).
== Leading digits of exponent ==
We don't know if it's possible to calculate the leading digits of either Skewes number, but we can compute the leading digits of their base-10 logarithms. The following transformation shows this (for \(Sk_1\)):
\(e^{e^{e^{79}}} = e^{10^{e^{79} \times log(e)}} = 10^{10^{e^{79} \times log(e)} \times log(e)} = 10^{10^{e^{79} \times log(e)+log(log(e))}}\). On the big number calculator: \(10^{e^{79} \times log(e)+log(log(e))} = 35,536,897,484,442,193,330...\), so \(Sk_1 = 10^{35,536,897,484,442,193,330...}\)
Also, similarly, \(Sk_2 = 10^{2,937,727,533,220,625,115,1...}\)
== Approximations in other notations ==
First Skewes number:
Second Skewes number:
==Sources==
==See also==
*googolduplex
*Graham's number
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ja:スキューズ数
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