Mills' constant

Last updated September 22, 2024

In number theory, Mills' constant is defined as the smallest positive real number A such that the floor function of the double exponential function

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.^[1] Its value is unproven, 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,\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 $a_{i-1}^{3}$ . 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 > $e^{e^{34}}$ , there is at least one prime between $a^{3}$ and $(a+1)^{3}$ .^[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 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.^[3]

As of 2024^[update], 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

(sequence A108739 in the OEIS ), 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.^[4] There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational.^[5]

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^[6] with value 2 (sequence A059784 in the OEIS ).

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.^[7]

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 $r>2.106\ldots$ .^[8] 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 $n$ , where $A\approx 1.00536773279814724017$ , and that $\lfloor B^{3^{13n}}\rfloor$ is prime for all positive integers $n$ , where $B\approx 3.8249998073439146171615551375$ .^[9]

Related Research Articles

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter $ζ$ (zeta), is a mathematical function of a complex variable defined as $for, and its analytic continuation elsewhere.$

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 3². The smallest positive square-free numbers are

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

In mathematics, the floor function is the function that takes as input a real number $x$ , and gives as output the greatest integer less than or equal to $x$ , denoted $⌊ x ⌋$ or $floor(x)$ . Similarly, the ceiling function maps $x$ to the smallest integer greater than or equal to $x$ , denoted $⌈ x ⌉$ or $ceil(x)$ .

In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:

<span class="mw-page-title-main">Prime-counting function</span> Function representing the number of primes less than or equal to a given number

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number $x$ . It is denoted by $π (x)$ (unrelated to the number $π$ ).

27 is the natural number following 26 and preceding 28.

48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens.

In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers.

<span class="mw-page-title-main">Mertens function</span> Summatory function of the Möbius function

In number theory, the Mertens function is defined for all positive integers n as

A powerful number is a positive integer m such that for every prime number p dividing m, p² also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a²b³, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.

In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however, they are computationally very slow. A number of constraints are known, showing what such a "formula" can and cannot be.

In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

Thomae's function is a real-valued function of a real variable that can be defined as:

The Copeland–Erdős constant is the concatenation of "0." with the base 10 representations of the prime numbers in order. Its value, using the modern definition of prime, is approximately

In mathematics, trailing zeros are a sequence of 0 in the decimal representation of a number, after which no other digits follow.

A double exponential function is a constant raised to the power of an exponential function. The general formula is $(where a >1 and b >1), which grows much more quickly than an exponential function. For example, if a = b = 10:$

In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.

In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius $. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name.$

In number theory, the totient summatory function $is a summatory function of Euler's totient function defined by:$

References

↑ 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 .
↑ 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.
↑ Caldwell, Chris (7 July 2006). "The Prime Database". Primes. 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. p. 5.4.1. MR 2165330.
↑ Finch, Steven R. (2003). "Mills' Constant". Mathematical Constants. Cambridge University Press. pp. 130–133. ISBN 0-521-81805-2.
↑ Warren Jr., Henry S. (2013). Hacker's Delight (2nd ed.). Addison-Wesley Professional. ISBN 9780321842688.
↑ Matomäki, K. (2010). "Prime-representing functions" (PDF). Acta Mathematica Hungarica . 128 (4): 307–314. doi: 10.1007/s10474-010-9191-x . S2CID 18960874.
↑ Tóth, László (2017). "A Variation on Mills-Like Prime-Representing Functions" (PDF). Journal of Integer Sequences. 20. p. 17.9.8. arXiv: 1801.08014 .
↑ 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.

External links

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

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Mills1947-1] 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 .

[Dudek2016-2] 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.

[Caldwell2006-3] Caldwell, Chris (7 July 2006). "The Prime Database". Primes. Retrieved 2017-05-11.

[CaldwellCheng2005-4] Caldwell, Chris K.; Cheng, Yuanyou (2005). "Determining Mills' Constant and a Note on Honaker's Problem". Journal of Integer Sequences. 8. p. 5.4.1. MR 2165330.

[Finch2003-5] Finch, Steven R. (2003). "Mills' Constant". Mathematical Constants. Cambridge University Press. pp. 130–133. ISBN 0-521-81805-2.

[WarrenJr2013-6] Warren Jr., Henry S. (2013). Hacker's Delight (2nd ed.). Addison-Wesley Professional. ISBN 9780321842688.

[Matomäki2010-7] Matomäki, K. (2010). "Prime-representing functions" (PDF). Acta Mathematica Hungarica . 128 (4): 307–314. doi: 10.1007/s10474-010-9191-x . S2CID 18960874.

[Tóth2017-8] Tóth, László (2017). "A Variation on Mills-Like Prime-Representing Functions" (PDF). Journal of Integer Sequences. 20. p. 17.9.8. arXiv: 1801.08014 .

[9] 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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]