Skewes’ number

Skewes’ number is the smallest number n for which π(n)>li(n), where π(x) is the prime counting function and li(x) is the logarithmic integralDlmfDlmfMathworldPlanetmathPlanetmath. The logarithmic integral is a good estimate for the prime counting function, but in the range of prime numbersMathworldPlanetmath for which we know all smaller primes, the logarithmic integral is an overestimate. Thus, Skewes’ number is the smallest number for which li(x) “goes from being an overestimate to being an underestimate.” (Wells, 1986)

The exact value of Skewes’ number is not currently known. Stanley Skewes in 1933 gave the lower bound eee79, approximately 10101034. He assumed the Riemann hypothesisMathworldPlanetmath to be true. Others have proven smaller bounds to as low as about 1.4×10316.

In the 1930s, Skewes’ number was the largest that had ever been used in a serious mathematical proof. It has since then been significantly dwarfed by Graham’s number. It still is the second largest number with its own entry in Wells’ The Penguin Dictionary of Curious and Interesting Numbers, appearing on the penultimate page of the main text.


  • 1 Bays, C. & Hudson, R. H. “A new bound for the smallest x with π(x)>li(x).” Math. Comput. 69 (2000): 1285 - 1296
  • 2 Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1986): 209
Title Skewes’ number
Canonical name SkewesNumber
Date of creation 2013-03-22 17:02:58
Last modified on 2013-03-22 17:02:58
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41
Synonym Skewes’s number
Synonym Skewes number