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| In [[number theory]], '''Mills' constant''' is defined as the smallest positive [[real number]] ''A'' such that the [[floor function|floor]] of the [[double exponential function]]
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| :<math> \lfloor A^{3^{n}} \rfloor</math>
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| is a [[prime number]], for all positive integers ''n''. This constant is named after [[William H. Mills]] who proved in 1947 the existence of ''A'' based on results of [[Guido Hoheisel]] and [[Albert Ingham]] on the [[prime gap]]s.
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| Its value is unknown, but if the [[Riemann hypothesis]] is true it is approximately 1.3063778838630806904686144926... {{OEIS|A051021}}.
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| ==Mills primes==
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| The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins
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| :2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... {{OEIS|A051254}}.
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| If ''a<sub>i</sub>'' denotes the ''i''<sup>th</sup> prime in this sequence, then ''a<sub>i</sub>'' can be calculated as the smallest prime number larger than <math>a_{i-1}^3</math>. In order to ensure that rounding <math>A^{3^n}</math>, for ''n'' = 1, 2, 3, …, produces this sequence of primes, it must be the case that <math>a_i < (a_{i-1}+1)^3</math>. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large [[cubic number]]s, which is sufficient to prove this inequality if we start from a sufficiently large first prime <math>a_1</math>. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and allowing the sequence of Mills' primes to begin at ''a''<sub>1</sub> = 2.
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| Currently, the largest known Mills prime (under the Riemann hypothesis) is
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| :<math>\displaystyle (((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220,</math>
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| which is 20,562 digits long.
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| ==Numerical calculation==
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| By calculating the sequence of Mills primes, one can approximate Mills' constant as
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| :<math>A\approx a(n)^{1/3^n}.</math>
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| {{harvtxt|Caldwell|Cheng|2005}} used this method to compute almost seven thousand base 10 digits of Mills' constant under the assumption that the [[Riemann hypothesis]] is true. There is no closed-form formula known for Mills' constant, and it is not even known whether this number is [[rational number|rational]] {{harv|Finch|2003}}.
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| ==See also==
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| *[[Formula for primes]]
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| ==References==
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| *{{citation|last1=Caldwell|first1=Chris K.|last2=Cheng|first2=Yuanyou|title=Determining Mills' Constant and a Note on Honaker's Problem|journal=Journal of Integer Sequences|volume=8|year=2005|issue=5.4.1|url=http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html}}.
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| *{{citation|first=Steven R.|last=Finch|title=Mathematical Constants|year=2003|publisher=Cambridge University Press|isbn=0-521-81805-2|contribution=Mills' Constant|pages=130–133|url=ftp://s208.math.msu.su/469000/dbcd69f8d83a96354dd49d21572c6432}}.
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| *{{citation|first=W. H.|last=Mills|title=A prime-representing function|journal=[[Bulletin of the American Mathematical Society]]|volume=53|year=1947|page=604|doi=10.1090/S0002-9904-1947-08849-2|issue=6|url = http://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf}}.
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| ==External links==
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| * {{MathWorld|urlname=MillsConstant|title=Mills' Constant}}
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| * [http://blogs.ethz.ch/kowalski/2009/04/02/who-remembers-the-mills-number/ Who remembers the Mills number?], E. Kowalski.
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| * [http://www.youtube.com/watch?v=6ltrPVPEwfo Awesome Prime Number Constant], Numberphile.
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| {{Prime number classes}}
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| [[Category:Mathematical constants]]
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| [[Category:Prime numbers]]
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