## Skewes' NumberA famous large number, commonly given as 10 ^{10^10^34}, that was
first derived in 1933 by the South African mathematician Samuel 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 John Conway
and Richard Guy^{3} have made the contradictory claim that the lower
bound is 10^{1167}. In any event, "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- Skewes, S. "On the difference pi(
*x*) - Li(*x*)." J. London Math. Soc., 8: 277-283 (1933). - te Riele, H. J. J. "On the Sign of the Difference pi(x) - Li(x)."
*Math. Comput*., 48: 323-328 (1987). - Conway, John Horton, and Guy, Richard K.
*The Book of Numbers*. New York: Springer-Verlag, 1996.
## Related category• NOTABLE NUMBERS | |||||

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