Definition:Skewes' Number

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 $\ds \int_2^n \frac {\d x} {\ln x}$.

Sources

• 1933: S. Skewes: On the difference $\map \pi x − \map \Li x$ (I) (J. London Math. Soc. Vol. 8, no. 4: pp. 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): $10^{10^{10^{34}}}$