Mills' constant

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

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

is a prime number, 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 and Albert Ingham on the prime gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... (sequence [[OEIS:{{#property:P829|from=Q1192957}}|{{#property:P829|from=Q1192957}}]] in 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, ... (sequence A051254 in 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 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 a₁ = 2.

As of 2015, the largest known Mills (probable) prime (under the Riemann hypothesis) is

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

(sequence A108739 in OEIS) which is 555,154 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 & Cheng (2005) used this method to compute almost seven thousand 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).

See also

• Formula for primes

References

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External links

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