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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 prime counting function and \(li(n)\) is the logarithmic integral. This bound was first proven assuming the Riemann hypothesis.[1] It's equal to \(e^{e^{e^{79}}} \approx 10^{10^{10^{34}}}\). 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 Edit

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 Edit

First Skewes number:

Notation Approximation
Scientific notation \(10^{10^{8.5*10^{33}}}\)
Up-arrow notation \(24\uparrow\uparrow4\)
Hyper-E notation \(\mathrm{E}34\#3\)
Fast-growing hierarchy \(f_2(f_2(f_2(108)))\)

Second Skewes number:

Notation Approximation
Scientific notation \(10^{10^{10^{963}}}\)
Up-arrow notation \(374\uparrow\uparrow4\)
Hyper-E notation \(\mathrm{E}963\#3\)
Fast-growing hierarchy \(f_2(f_2(f_2(3189)))\)

Sources Edit

  1. Skewes Number

See also Edit