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. 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 $$1.4 \cdot 10^{316}$$.

## Contents

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:

Notation Approximation
Scientific notation $$10^{10^{8.5*10^{33}}}$$
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 $$10^{10^{10^{963}}}$$
Arrow notation $$374\uparrow\uparrow4$$
Hyper-E notation $$\mathrm{E}963\#3$$
Fast-growing hierarchy $$f_2(f_2(f_2(3189)))$$