Innumber theory,Mills' constant is defined as the smallest positivereal numberA such that thefloor function of thedouble exponential function

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

is aprime number for all positivenatural numbersn. This constant is named afterWilliam Harold Mills who proved in 1947 the existence ofA based on results ofGuido Hoheisel andAlbert Ingham on theprime gaps.^[1] Its value is unproven, but if theRiemann hypothesis is true, it is approximately 1.3063778838630806904686144926... (sequenceA051021 in theOEIS).

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

${\displaystyle 2,11,1361,2521008887,16022236204009818131831320183,}$

${\displaystyle 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\dots }$ (sequenceA051254 in theOEIS).

If${\displaystyle a_i}$ denotes theith prime in this sequence, then${\displaystyle a_i}$ can be calculated as the smallest prime number larger than${\displaystyle a_{i-1}^3}$. In order to ensure that rounding${\displaystyle A^{3^n}}$, forn = 1, 2, 3, ..., produces this sequence of primes, it must be the case that. The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently largecube 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 thesufficiently large condition to be removed, and allowing the sequence of Mills primes to begin ata₁ = 2.

For all${\displaystyle a_i>e^{e^{32.537}}}$, there is at least one prime between${\displaystyle a_i^3}$ and${\displaystyle (a_i+1{)}^3}$.^[2] 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 beenproved 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.^[3]

As of 2024^[update], the largest known Millsprobable prime (under the Riemann hypothesis) is

(sequenceA108739 in theOEIS), which is 1,665,461 digits long.

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

${\displaystyle A\approx a_n^{1/3^n}.}$

Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that theRiemann hypothesis is true.^[4] Mills' constant is not known to have aclosed-form formula,^[5] but it is known to beirrational.^[6]

There is nothing special about the middle exponent value of 3. It is possible to produce similar prime-generatingfunctions for different middle exponent values. In fact, for any real number above 2.106..., it is possible to find a different constantA that will work with this middle exponent to always produce primes. Moreover, ifLegendre's conjecture is true, the middle exponent can be replaced^[7] with value 2 (sequenceA059784 in theOEIS).

Matomäki showed unconditionally (without assuming Legendre's conjecture) the existence of a (possibly large) constantA such that${\displaystyle \lfloor A^{2^n}\rfloor }$ is prime for alln.^[8]

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

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

is also prime-representing for${\displaystyle r>2.106,r\in N\dots }$.^[9]In the case${\displaystyle r=3}$, the value of the constant${\displaystyle B}$ begins with 1.24055470525201424067... The first few primes generated are:

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

Without assuming the Riemann hypothesis, Elsholtz proved that${\displaystyle \lfloor A^{{10}^{10n}}\rfloor }$ is prime for all positive integersn, where${\displaystyle A\approx 1.00536773279814724017}$, and that${\displaystyle \lfloor B^{3^{13n}}\rfloor }$ is prime for all positive integersn, where${\displaystyle B\approx 3.8249998073439146171615551375}$.^[10]

Duc-Son Tran proved that in Tóth's result above, the number r can be replaced by any real number${\displaystyle >8/3}$.Moreover, he demonstrated another result stating that for any given real number B>1, there exists a number r such that${\displaystyle \lceil B^{r^n}\rceil }$is also prime-representing.^[11]

• Formula for primes

• Cheng, Yuanyou Furui 2010 (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.{{cite journal}}: CS1 maint: numeric names: authors list (link)

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

• Mathematical constants
• Prime numbers

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