Mills (1947) proved the existence of a real constant
prime for all integers , where is the floor function.
Mills (1947) did not, however, determine , or even a range for .
A generalization of Mills' theorem to an arbitrary sequence of
integers is given as an exercise by Ellison and Ellison (1985).
is prime for all integers is known as Mills' constant.
Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let
be the th prime, then there exists
This has more recently been strengthened to
(Mozzochi 1986). If the
is true, then Cramér (1937) showed that
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 is known, the primes themselves
must be known in advance to determine .
See also Mills' Constant
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. New York: Wiley, pp. 31-32, 1985. Prime
Numbers. Finch, S. R.
"Mills' Constant." §2.13 in Cambridge, England: Cambridge University Press, pp. 130-133,
Constants. Hardy, G. H. and Wright, E. M. Oxford, England: Clarendon
Press, 1979. An
Introduction to the Theory of Numbers, 5th ed. 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. New York: Wiley, p. 65, 1951. Introduction
to Number Theory. Ribenboim,
P. New York: Springer-Verlag, pp. 186-187,
New Book of Prime Number Records. Ribenboim, P. New York: Springer-Verlag, pp. 109-110, 1991. The
Little Book of Big Primes. Referenced
on Wolfram|Alpha Mills' Theorem
Cite this as:
Weisstein, Eric W. "Mills' Theorem." From
--A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/MillsTheorem.html