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

: $\left\lfloor A^{3^{n}} \right\rfloor$

is a prime number for all positive natural numbers n. This constant is named after William Harold Mills who proved in 1947 the existence of A based on results of Guido Hoheisel and Albert Ingham on the prime gaps.{{r|Mills1947}} Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... {{OEIS|id=A051021|name=Decimal expansion of Mills's constant, assuming the Riemann Hypothesis is true}}.

Mills primes

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

: $2, 11, 1361, 2521008887, 16022236204009818131831320183,$

: $4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, \ldots$ {{OEIS|A051254}}.

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 cube 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.

For all a > $e^{e^{32.537}}$ , there is at least one prime between $a^3$ and $(a+1)^3$ .{{r|Cully-Hugill2023}} 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

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

and has 20562 digits.{{r|Caldwell2006}}

{{As of|2024}}, the largest known Mills probable prime (under the Riemann hypothesis) is

: $\displaystyle (((((((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840)^3+1623568)^3+8436308$

{{OEIS|A108739}}, which is 1,665,461 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 and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that the Riemann hypothesis is true.{{r|CaldwellCheng2005}} Mills' constant is not known to have a closed-form formula,{{r|Finch2003}} but it is known to be irrational.{{Cite journal |last=Saito |first=Kota |date=2025 |title=Mills' constant is irrational |url=https://onlinelibrary.wiley.com/doi/abs/10.1112/mtk.70027 |journal=Mathematika |language=en |volume=71 |issue=3 |pages=e70027 |arxiv=2404.19461 |doi=10.1112/mtk.70027 |issn=2041-7942}}

Generalisations

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{{r|WarrenJr2013}} with value 2 {{OEIS|A059784}}.

Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constant A such that $\lfloor A^{2^{n}} \rfloor$ is prime for all n.{{r|Matomäki2010}}

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

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

is also prime-representing for $r>2.106\ldots$ .{{r|Tóth2017}}

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

: $2, 7, 337, 38272739, 56062005704198360319209, 176199995814327287356671209104585864397055039072110696028654438846269, \ldots$

Without assuming the Riemann hypothesis, Elsholtz proved that $\lfloor A^{10^{10n}} \rfloor$ is prime for all positive integers {{mvar|n}}, where $A \approx 1.00536773279814724017$ , and that $\lfloor B^{3^{13n}} \rfloor$ is prime for all positive integers {{mvar|n}}, where $B \approx 3.8249998073439146171615551375$ .{{cite journal |last=Elsholtz|first=Christian |title=Unconditional Prime-Representing Functions, Following Mills |journal=American Mathematical Monthly|volume=127|issue=7|year=2020|pages=639–642 |doi=10.1080/00029890.2020.1751560 |arxiv = 2004.01285|s2cid=214795216 }}

References

{{reflist|refs=

{{cite journal |last=Mills |first=W. H. |date=1947 |title=A prime-representing function |journal=Bulletin of the American Mathematical Society |volume=53 |issue=6 |page=604 |doi=10.1090/S0002-9904-1947-08849-2 |doi-access=free |url=https://www.ams.org/journals/bull/1947-53-06/S0002-9904-1947-08849-2/S0002-9904-1947-08849-2.pdf }}

{{cite thesis |degree=Ph.D. |title=Explicit estimates for the distribution of primes |last=Cully-Hugill |first=Michaela |year=2023 |publisher=UNSW Canberra |doi=10.26190/unsworks/24916}}

{{cite web |last=Caldwell |first=Chris |date=7 July 2006 |title=The Prime Database |website=Primes |url=http://primes.utm.edu/primes/page.php?id=77907 |access-date=2017-05-11}}

{{cite journal |last1=Caldwell |first1=Chris K. |last2=Cheng |first2=Yuanyou |date=2005 |title=Determining Mills' Constant and a Note on Honaker's Problem |journal=Journal of Integer Sequences |volume=8 |at=p. 5.4.1 |url=http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html |mr=2165330 }}

{{cite book |last=Finch |first=Steven R. |title=Mathematical Constants |date=2003 |publisher=Cambridge University Press |isbn=0-521-81805-2 |contribution=Mills' Constant |at=[https://archive.org/details/mathematicalcons0000finc/page/130 pp. 130–133] }}

{{cite book |last=Warren Jr. |first=Henry S. |date=2013 |title=Hacker's Delight |edition=2nd |publisher=Addison-Wesley Professional |isbn=9780321842688 }}

{{cite journal |last=Matomäki |first=K. |author-link=Kaisa Matomäki |date=2010 |title=Prime-representing functions |journal=Acta Mathematica Hungarica |volume=128 |issue=4 |pages=307–314 |doi=10.1007/s10474-010-9191-x | doi-access=free |s2cid=18960874 |url=http://users.utu.fi/ksmato/papers/Primerepfunc.pdf }}

{{cite journal |last=Tóth |first=László |date=2017 |title=A Variation on Mills-Like Prime-Representing Functions |journal=Journal of Integer Sequences |volume=20 |at=p. 17.9.8 |arxiv=1801.08014 |url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth2/toth32.pdf }}

}}

External links

{{MathWorld|urlname=MillsConstant|title=Mills' Constant}}
[http://blogs.ethz.ch/kowalski/2009/04/02/who-remembers-the-mills-number/ Who remembers the Mills number?], E. Kowalski.
[https://www.youtube.com/watch?v=6ltrPVPEwfo Awesome Prime Number Constant], Numberphile.

Category:Mathematical constants

Category:Prime numbers