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 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 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 Riele² reduced dramatically the upper bound of the first crossing to e^{e^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: 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.