Mills' constant

In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function

\lfloor A^{3^{n}} \rfloor

is a prime number, for all natural numbers 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 gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... (sequence A051021 in the OEIS).

Mills primes

The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... (sequence A051254 in the OEIS).

If a_i denotes the i^th prime in this sequence, then a_i can be calculated as the smallest prime number larger than $a_{{i-1}}^{3}$ . In order to ensure that rounding $A^{{3^{n}}}$ , for n = 1, 2, 3, …, produces this sequence of primes, it must be the case that $a_{i}<(a_{{i-1}}+1)^{3}$ . The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large cubic numbers, which is sufficient to prove this inequality if we start from a sufficiently large first prime $a_{1}$ . 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 a₁ = 2.

As of April 2017, the 11^th number in the sequence is the largest one that has been proved prime. It is

\displaystyle (((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220

and has 20562 digits.^[1]

As of 2015, the largest known Mills (probable) prime (under the Riemann hypothesis) is^[2]

\displaystyle ((((((((((((2^{3}+3)^{3}+30)^{3}+6)^{3}+80)^{3}+12)^{3}+450)^{3}+894)^{3}+3636)^{3}+70756)^{3}+97220)^{3}+66768)^{3}+300840)^{3}+1623568

(sequence A108739 in the OEIS), which is 555,154 digits long.

Numerical calculation

By calculating the sequence of Mills primes, one can approximate Mills' constant as

A\approx a(n)^{{1/3^{n}}}.

Caldwell & Cheng (2005) used this method to compute 6850 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 (Finch 2003).

Middle exponent

There is nothing special about the middle exponent value of 3. It is possible to produce similar prime-generating functions for different middle exponent values. In fact, for any real number above 2.106, it is possible to find a different constant A that will work with this middle exponent to always produce primes. Moreover, if Legendre's conjecture is true, the middle exponent can be replaced with value 2 (Warren Jr. 2013) (sequence A059784 in the OEIS).

Fractional representations

Below are fractions which approximate Mills' constant, listed in order of increasing accuracy (with continued-fraction convergents in bold^[3]):

1/1, 3/2, 4/3, 9/7, 13/10, 17/13, 47/36, 64/49, 81/62, 145/111, 226/173, 307/235, 840/643, 1147/878, 3134/2399, 4281/3277, 5428/4155, 6575/5033, 12003/9188, 221482/169539, 233485/178727, 245488/187915, 257491/197103, 269494/206291, 281497/215479, 293500/224667, 305503/233855, 317506/243043, 329509/252231, 341512/261419, 353515/270607, 365518/279795, 377521/288983, 389524/298171, 401527/307359, 413530/316547, 425533/325735, 4692866/3592273, 5118399/3918008, 5543932/4243743, 5969465/4569478, 6394998/4895213, 6820531/5220948, 7246064/5546683,7671597/5872418, 8097130/6198153, 8522663/6523888, 8948196/6849623, 9373729/7175358, 27695654/21200339, 37069383/28375697, 46443112/35551055, 148703065/113828523, 195146177/149379578, 241589289/184930633, 436735466/334310211, 1115060221/853551055, 1551795687/1187861266, 1988531153/1522171477, 3540326840/2710032743, 33414737247/25578155953, ...

References

↑ Caldwell, Chris (2006-07-07). "The Prime Database". Retrieved 2017-05-11.
↑ PRP records
↑ A123561, The On-Line Encyclopedia of Integer Sequences

Caldwell, Chris K.; Cheng, Yuanyou (2005), "Determining Mills' Constant and a Note on Honaker's Problem", Journal of Integer Sequences, 8 (5.4.1), MR 2165330 .
Finch, Steven R. (2003), "Mills' Constant", Mathematical Constants, Cambridge University Press, pp. 130–133, ISBN 0-521-81805-2 .
Mills, W. H. (1947), "A prime-representing function" (PDF), Bulletin of the American Mathematical Society, 53 (6): 604, doi:10.1090/S0002-9904-1947-08849-2 .
Warren Jr., Henry S. (2013), Hacker's Delight. 2nd edition, Addison-Wesley Professional, ISBN 978-0-321-84268-8 .

External links

Weisstein, Eric W. "Mills' Constant". MathWorld.

Who remembers the Mills number?, E. Kowalski.
Awesome Prime Number Constant, Numberphile.

Prime number classes
By formula	Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1) Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^u·3^v + 1) Quartan (x⁴ + y⁴) Solinas (2^a ± 2^b ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y) Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Mills (floor(A^3ⁿ))
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit (10ⁿ − 1)/9 Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain (p, 2p + 1) Cunningham chain (p, 2p ± 1, …) Safe (p, (p − 1)/2) Arithmetic progression (p + a·n, n = 0, 1, …) Balanced (consecutive p − n, p, p + n)
By size	Titanic (1,000+ digits) Gigantic (10,000+) Mega (1,000,000+) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Almost prime Semiprime Interprime
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 50 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229
List of prime numbers

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