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# Skewes' Number

Professor Stanley Skewes.

Skewes' Number is a famous large number, commonly given as 1010^10^34, 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 1022, 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 ee^e^79. This is called the first or Riemann true Skewes' Number. Converted to base 10, the value can be approximated as 1010^10^34, or more accurately as 1010^8.852142×10^33 or 1010^8852142197543270606106100452735038.55.

In 1987, the Dutch mathematician Herman te Riele2 reduced dramatically the upper bound of the first crossing to ee^27/4, or approximately 8.185 × 10370, while Bays and Hudson3 lowered the upper bound is 10316. In any event, the original "Skewes' Number" is now only of historical interest.

Skewes also defined the limit if the Riemann Hypothesis is false: 1010^10^1000. 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.