Mills' constant

# Mills' constant

Discussion

Encyclopedia
In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, Mills' constant is defined as the smallest positive real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

A such that the floor
Floor function
In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...

of the double exponential function

is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, for all positive integers n. This constant is named after William H. Mills who proved in 1947 the existence of A based on results of Guido Hoheisel
Guido Hoheisel
Guido Hoheisel was a German mathematician, a professor of mathematics at the University of Cologne. He did his PhD in 1920 from the University of Berlin under the supervision of Erhard Schmidt....

and Albert Ingham
Albert Ingham
Albert Edward Ingham was an English mathematician.Ingham was born in Northampton. He went to Stafford Grammar School and Trinity College, Cambridge . He obtained his Ph.D., which was supervised by John Edensor Littlewood, from the University of Cambridge. He supervised the Ph.D.s of C. Brian...

on the prime gaps.
Its value is unknown, but if the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

is true it is approximately 1.3063778838630806904686144926... .

## 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... .

If a(i) denotes the ith 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 A3n, 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 cubic 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 a(1) = 2.

Currently, the largest known Mills prime (under the Riemann hypothesis) is
which is 20,562 digits long.

## Numerical calculation

By calculating the sequence of Mills primes, one can approximate Mills' constant as
used this method to compute almost seven thousand base 10 digits of Mills' constant under the assumption that the Riemann hypothesis
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

is true. There is no closed-form formula known for Mills' constant, and it is not even known whether this number is rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

.