Definition:Skewes' Number/Historical Note

Historical Note on Skewes' Number

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

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $10^{10^{10^{34}}}$
• 1989: Paulo Ribenboim: The Book of Prime Number Records (2nd ed.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $10^{10^{10^{34}}}$