# Skewes' number

Professor Stanley Skewes.

Skewes' number is a famous large number, commonly given as 10^{10^10^34}, that was
first derived in 1933 by the South African mathematician Stanley Skewes in
a proof involving prime numbers.^{1} G. H. Hardy once described Skewes' number
as "the largest number which has ever served any definite purpose in mathematics,"
though it has long since lost that distinction.

Skewes' numbers – there are actually two of them – came about
from a study of the frequency with which prime numbers occur. Gauss's well-known
estimate of the number of prime numbers less than or equal to *n*,
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 or **Riemann true 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^{10^8852142197543270606106100452735038.55}.

In 1987, the Dutch mathematician Herman te Riele^{2} reduced dramatically
the upper bound of the first crossing to e^{e^27/4}, or approximately
8.185 × 10^{370}, while Bays and Hudson^{3} lowered the upper
bound is 10^{316}. In any event, the original "Skewes' Number" is now only of
historical interest.

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

### References

1. Skewes, S. (1933), "On the difference pi(*x*) - Li(*x*)." J.
London Math. Soc., 8: 277–283 (1933).

2. te Riele, H. J. J. (1987), "On the Sign of the Difference pi(x) - Li(x)." *Math. Comput*., 48: 323–328.

3. Bays, C. and Hudson, R. H. (2000), "A new bound for the smallest x with π(x) > li(x)", Mathematics of Computation, 69 (231): 1285–1296.