Skewes's number

In number theory, Skewes's number is the smallest natural number ${\displaystyle x}$ for which the prime-counting function ${\displaystyle \pi (x)}$ exceeds the logarithmic integral function ${\displaystyle \operatorname {li} (x).}$ It is named for the South African mathematician Stanley Skewes who first computed an upper bound on its value.

The exact value of Skewes's number is still not known, but it is known that there is a crossing between ${\displaystyle \pi (x)<\operatorname {li} (x)}$ and ${\displaystyle \pi (x)>\operatorname {li} (x)}$ near ${\displaystyle e^{727.95133}<1.397\times 10^{316}.}$ It is not known whether this is the smallest crossing.

The name is sometimes also applied to either of the large number bounds which Skewes found.