Mills' Constant

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Mills' theorem states that there exists a real constant A such that |_A^(3^n)_| is prime for all positive integers n (Mills 1947). While for each value of c>=2.106, there are uncountably many possible values of A such that |_A^(c^n)_| is prime for all positive integers n (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least theta such that

is prime for all positive integers n, giving a value of

(OEIS A051021).

f_(n+1) is therefore given by the next prime after f_n^3, and the values of f_n are known as Mills' primes (Caldwell and Cheng 2005).

Caldwell and Cheng (2005) computed more than 6850 digits of theta assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately 185000 digits are now known.

It is not known if theta is irrational.

See also

Floor Function, Mills' Prime, Mills' Theorem, Power Floor Prime Sequence, Prime Formulas, Prime Number

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References

Caldwell, C. K. and Cheng, Y. "Determining Mills' Constant and a Note on Honaker's Problem." J. Integer Sequences 8, Article 05.4.1, 1-9, 2005. https://cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html.Finch, S. R. "Mills' Constant." §2.13 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 130-133, 2003.Mills, W. H. "A Prime-Representing Function." Bull. Amer. Math. Soc. 53, 604, 1947.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 109-110, 1991.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 186-187, 1996.Sloane, N. J. A. Sequences A051021, A051254, and A108739 in "The On-Line Encyclopedia of Integer Sequences."

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Mills' Constant

Cite this as:

Weisstein, Eric W. "Mills' Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MillsConstant.html

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