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 2015, the largest known Mills (probable) prime (under the Riemann hypothesis) is^[1]

\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 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 (Finch 2003).

Accurate Fractional Representations

Below are fractions which approximate Mills' Constant. They are listed in order of increasing accuracy. 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

↑ PRP records

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 .

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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