Mills' Constant
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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

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

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 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

${\displaystyle 2,11,1361,2521008887,16022236204009818131831320183,4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\ldots }$ (sequence in the OEIS).

If ai denotes the ith prime in this sequence, then ai can be calculated as the smallest prime number larger than ${\displaystyle a_{i-1}^{3}}$. In order to ensure that rounding ${\displaystyle A^{3^{n}}}$, for n = 1, 2, 3, …, produces this sequence of primes, it must be the case that ${\displaystyle a_{i}<(a_{i-1}+1)^{3}}$. 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 ${\displaystyle a_{1}}$. 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.

For all a > ${\displaystyle e^{e^{15}}}$, there is at least one prime between ${\displaystyle a^{3}}$ and ${\displaystyle (a+1)^{3}}$ (Cheng 2010). This number has 1,419,717 decimal digits, so checking every number under this figure would be infeasible. However, the value of Mills' constant can be verified by calculating the first prime in the sequence that is greater than that figure.

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

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

and has 20562 digits (Caldwell 2006).

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

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

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

## Numerical calculation

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

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

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).

## Fractional representations

Below are fractions which approximate Mills' constant, listed in order of increasing accuracy (with continued-fraction convergents in bold) (sequence in the OEIS):

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, ...

## 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 in the OEIS).

## Floor and ceiling functions

There is nothing special either about the floor function in the formula. Tóth proved that the function defined by

${\displaystyle \lceil A^{r^{n}}\rceil }$

is also prime-representing for ${\displaystyle r>2.106\ldots }$ (Tóth 2017).

In the case ${\displaystyle r=3}$, the value of the constant ${\displaystyle A}$ begins with 1.24055470525201424067... The first few primes generated are:

${\displaystyle 2,7,337,38272739,56062005704198360319209,176199995814327287356671209104585864397055039072110696028654438846269,\ldots }$

## References

• Caldwell, Chris (2006-07-07), The Prime Database, retrieved
• 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.
• Cheng, Yuan-You Fu-Rui (2010), "Explicit estimate on primes between consecutive cubes", The Rocky Mountain Journal of Mathematics, 40 (1): 117-153, arXiv:0810.2113, doi:10.1216/RMJ-2010-40-1-117, MR 2607111
• 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.
• Tóth, László (2017), "A Variation on Mills-Like Prime-Representing Functions" (PDF), Journal of Integer Sequences, 20 (17.9.8).