米尔斯常数

米尔斯常数是使对于所有正整数n，二重指数函数 A³ⁿ

的整数部分都是素数的最小正实数A。这个常数以W·H·米尔斯命名，他在1947年证明了这个常数的存在。

米尔斯常数的值是未知的，但如果黎曼猜想成立，它的值大约为： A≈ 1.30637788386308069046... （A051021 （页面存档备份，存于互联网档案馆））。

米尔斯素数

由米尔斯常数所产生的素数称为米尔斯素数；如果黎曼猜想成立，这个数列的最初几项为：

2, 11, 1361, 2521008887…… （OEIS数列A051254）。

如果用a(i)来表示数列中的第i个素数，则a(i)可以计算为大于a(i −1)3的最小的素数。为了保证当n = 1，2，3，……时，A3n的整数部分是这个素数数列，必须有a(i) < (a(i −1) + 1)3。Hoheisel和Ingham的结果保证了在任何两个足够大的立方数之间一定有一个素数，这足以证明这个不等式，如果我们从一个足够大的素数a(1)开始。从黎曼猜想，可以推出任何两个连续的立方数之间一定有一个素数，这样就可以去掉足够大的条件，并允许米尔斯素数的数列从a(1) = 2开始。

目前已知最大的米尔斯素数（假设黎曼猜想成立）是： (((((((((2³+3)³+30)³+6)³+80)³+12)³+450)³+894)³+3636)³+70756)³+97220,

它有20,562位。

计算

通过计算米尔斯素数，我们可以近似计算米尔斯常数为： A≈ a(n)¹/3ⁿ.

Caldwell & Cheng (2005)用这个方法计算出米尔斯常数的差不多七千位数。目前还没有闭合公式可以计算米尔斯常数，甚至不知道它是不是有理数(Finch 2003)。

In number theory, Mills' constant is defined as the smallest positive real number A such that （公式略）

is a prime number for all positive natural numbers n (where （公式略） denotes the floor function). This constant is named after William Harold Mills who in 1947 proved the existence of A based on results of Guido Hoheisel and Albert Ingham on prime gaps. 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 （公式略）

(sequence A051254 in the OEIS). If aᵢ denotes the i th prime in this sequence, then aᵢ can be calculated as the smallest prime number larger than aᵢ₋₁³. In order to ensure that rounding A³ⁿ for all n≥ 1 produces this sequence of primes, it must be the case that aᵢ<(aᵢ₋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 a₁. 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 aᵢ>e^(e)^(32.537), there is at least one prime between aᵢ³ and (aᵢ+1)³. 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 (((((((((2³+3)³+30)³+6)³+80)³+12)³+450)³+894)³+3636)³+70756)³+97220

and has 20562 digits.

As of 2024, the largest known Mills probable prime (under the Riemann hypothesis) is （公式略）

(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≈ aₙ¹/3ⁿ.

Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills' constant under the assumption that the Riemann hypothesis is true. Mills' constant is not known to have a closed-form formula, but it is known to be irrational. In addition, it is either transcendental (which would be implied by Cramér's conjecture) or the 3ⁿ -th root of a cubic Pisot–Vijayaraghavan number.

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 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 （公式略） is prime for all n. In contrast to Mills' constant, it is unknown if the smallest such A is irrational; the analogous constant where the middle exponent is an integer ≥ 4 is always transcendental.

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 （公式略）

is also prime-representing for r>2.106, r∈ N…

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

Without assuming the Riemann hypothesis, Elsholtz proved that （公式略） is prime for all positive integers n, where A≈ 1.00536773279814724017, and that （公式略） is prime for all positive integers n, where B≈ 3.8249998073439146171615551375.

Duc-Son Tran proved that in Tóth's result above, the number r can be replaced by any real number >8/3.

Moreover, he demonstrated another result stating that for any given real number B>1, there exists a number r such that （公式略）

is also prime-representing.