Simply put, what is Skewes Number?

Answer 1

Skewes' numbers – there are actually two of them – came about from a study of the frequency with which prime numbers occur.

The number of prime numbers less than or equal to n is denoted pi(n).

Gauss's well-known estimate of pi(n) is the integral from u=0 to u=n of 1/(log u); this integral is called Li(n).

In 1914 the English mathematician John Littlewood proved that pi(x) - Li(x) assumes both positive and negative values infinitely often. For all values of n up to 10^22, which is as far as computations have gone so far, Li(n) has turned out to be an overestimate. But Littlewood's result showed that above some value of n it becomes an underestimate, then at an even higher value of n it becomes an overestimate again, and so on.

This is where Skewes' Number comes in. Skewes showed that, if the Riemann Hypothesis is true, the first crossing can't be greater than e^e^e^79. This is called the First Skewes' Number. Converted to base 10, the value can be approximated as 10^10^10^34, or more accurately as 10^10^8.852142×10^33 or 10^10^8852142197543270606106100452735039

Skewes also defined the limit if the Riemann Hypothesis is false: 10^10^10^1000. This is known as the second Skewes' Number.