Mills' constant

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In mathematics, Mills' constant is defined as the smallest positive real number A; such that the integer part

is a prime number, for all positive integers n. Its value is approximately

(Sloane's A051021).

In 2005, Chris Caldwell and Ying Cheng computed almost seven thousand base 10 digits of Mills' constant under the assumption that the Riemann hypothesis is true. It is hard to calculate Mills' constant accurately (although several thousand digits are known), because to do that with the current state of knowledge one needs to know the primes it generates. There is no formula for this number as of now, and it is not even known if this number is rational. It is known, however, that the first few primes generated (the so-called Mills primes) are 2, 11, 1361, 2521008887... (sequence A051254 in OEIS).

This constant is named after W. H. Mills who proved in 1947 the existence of θ based on results of Hoheisel and A. E. Ingham on the gaps between the primes.

[edit] See also

• Formula for primes

[edit] References

• Caldwell, Chris K. and Cheng, Yuanyou. "Determining Mills' Constant and a Note on Honaker's Problem." Journal of Integer Sequences Vol. 8 (2005), article 05.4.1. http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html

[edit] External link

• MathWorld page