Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let be the th prime, then there exists
a constant
such that
(2)
for all .
This has more recently been strengthened to
(3)
(Mozzochi 1986). If the Riemann hypothesis
is true, then Cramér (1937) showed that
(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 is known, the primes themselves
must be known in advance to determine .
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C. K. and Cheng, Y. "Determining Mills' Constant and a Note on Honaker's
Problem." J. Integer Sequences8, 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
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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."
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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,
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