## FANDOM

10,827 Pages

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 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 $$10^{14}$$ 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: $$10^{e^{79} \times log(e)+log(log(e))} = 35536897484442193330...$$, so $$Sk_1 = 10^{35536897484442193330...}$$

Also, similarly, $$Sk_2 = 10^{29377275332206251151...}$$

## Sources Edit

1. Skewes Number