Mills' constant

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 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 ${\displaystyle a_{i-1}^{3}}$. 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 cube 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 > ${\displaystyle e^{e^{34}}}$, there is at least one prime between ${\displaystyle a^{3}}$ and ${\displaystyle (a+1)^{3}}$ (Dudek 2016). This upper bound is much too large to be practical, as it is infeasible to check every number below that figure. 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 A108739 in the OEIS), which is 555,154 digits long.

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). If it is rational, and if we can calculate its decimal expansion to the point where it repeats, this will allow us to generate infinitely many provable primes.

Below are fractions which approximate Mills' constant, listed in order of increasing accuracy (with continued-fraction convergents in bold) (sequence A123561 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, ...

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).

Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constant A such that ${\displaystyle \lfloor A^{2^{n}}\rfloor }$ is prime for all n (Matomäki 2010).

Additionally, Tóth proved that the floor function in the formula could be replaced with the ceiling function, so that there exists a constant ${\displaystyle B}$ such that

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

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

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

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

• Formula for primes

• Caldwell, Chris (2006-07-07), The Prime Database, retrieved 2017-05-11
• 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, S2CID 15502941
• Dudek, Adrian W. (2016), "An explicit result for primes between cubes", Functiones et Approximatio Commentarii Mathematici, 55 (2): 177–197, arXiv:1401.4233, doi:10.7169/facm/2016.55.2.3, MR 3584567, S2CID 119143089
• Elsholtz, Christian (2020), "Unconditional Prime-Representing Functions, Following Mills", American Mathematical Monthly, 127 (7): 639–642, arXiv:2004.01285, doi:10.1080/00029890.2020.1751560, S2CID 214795216.
• Finch, Steven R. (2003), "Mills' Constant", Mathematical Constants, Cambridge University Press, pp. 130–133, ISBN 0-521-81805-2.
• Matomäki, K. (2010), "Prime-representing functions" (PDF), Acta Mathematica Hungarica, 128 (4): 307–314, doi:10.1007/s10474-010-9191-x, S2CID 18960874
• 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.
• Tóth, László (2017), "A Variation on Mills-Like Prime-Representing Functions" (PDF), Journal of Integer Sequences, 20: 17.9.8, arXiv:1801.08014.
• Warren Jr., Henry S. (2013), Hacker's Delight (2nd ed.), Addison-Wesley Professional, ISBN 978-0-321-84268-8.

• Weisstein, Eric W. "Mills' Constant". MathWorld.
• Who remembers the Mills number?, E. Kowalski.
• Awesome Prime Number Constant, Numberphile.