# Definition:Skewes' Number

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## Definition

**Skewes' number** is:

- $10^{10^{10^{34} } }$

In Knuth notation this can be presented as:

- $10 \uparrow \paren {10 \uparrow \paren {10 \uparrow 34} }$

## Also known as

The name can also be seen presented as:

**Skewes Number****Skewes's Number**

## Source of Name

This entry was named for Stanley Skewes.

## Historical Note

Stanley Skewes deduced the number which now bears his name in $1933$.

Godfrey Harold Hardy referred to it as:

*the largest number which has ever served any definite purpose in mathematics.*

By way of comparison, the number of particles in the universe has been estimated at somewhere between $10^{80}$ and $10^{87}$.

However, Skewes' estimate has been reduced somewhat more recently.

For example, Hermanus Johannes Joseph te Riele has shown that there are many $n$ between $6 \cdotp 62 \times 10^{370}$ and $6 \cdotp 69 \times 10^{370}$ for which $\map \pi n$ is less than $\displaystyle \int_2^n \frac {\d x} {\ln x}$.

## Sources

- 1933: S. Skewes:
*On the difference $\map \pi x − \map \Li x$ (I)*(*J. London Math. Soc.***Vol. 8**,*no. 4*: 277 – 283)

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $10^{10^{10^{34}}}$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10^{10^{10^{34}}}$