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Mills' Theorem

Mills (1947) proved the existence of a real constant A such that

|_A^(3^n)_|
(1)

is prime for all integers n>=1, where |_x_| is the floor function. Mills (1947) did not, however, determine A, or even a range for A.

A generalization of Mills' theorem to an arbitrary sequence of positive integers is given as an exercise by Ellison and Ellison (1985).

The least theta such that |_theta^(3^n)_| is prime for all integers n>=1 is known as Mills' constant.

Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let p_n be the nth prime, then there exists a constant K such that

p_(n+1)-p_n<Kp_n^(5/8)
(2)

for all n. This has more recently been strengthened to

p_(n+1)-p_n<Kp_n^(1051/1920)
(3)

(Mozzochi 1986). If the Riemann hypothesis is true, then Cramér (1937) showed that

p_(n+1)-p_n=O(lnp_nsqrt(p_n))
(4)

(Finch 2003).

Hardy and Wright (1979) and Ribenboim (1996) point out that, despite the beauty of such prime formulas, they do not have any practical consequences. In fact, unless the exact value of theta is known, the primes themselves must be known in advance to determine theta.

See also

Mills' Constant, Power Floor Prime Sequence

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References

Caldwell, C. "Mills' Theorem--A Generalization." http://www.utm.edu/research/primes/notes/proofs/A3n.html.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. http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html.Ellison, W. and Ellison, F. Prime Numbers. New York: Wiley, pp. 31-32, 1985.Finch, S. R. "Mills' Constant." §2.13 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 130-133, 2003.Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.Hoheisel, G. "Primzahlprobleme in der Analysis." Sitzungsber. der Preuss. Akad. Wissensch. 2, 580-588, 1930.Ingham, A. E. "On the Difference Between Consecutive Primes." Quart. J. Math. 8, 255-266, 1937.Mills, W. H. "A Prime-Representing Function." Bull. Amer. Math. Soc. 53, 604, 1947.Mozzochi, C. J. "On the Difference Between Consecutive Primes." J. Number Th. 24, 181-187, 1986.Nagell, T. Introduction to Number Theory. New York: Wiley, p. 65, 1951.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 186-187, 1996.Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 109-110, 1991.

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Mills' Theorem

Cite this as:

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

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