In number theory, Mills' constant is defined as the smallest positive real number A {\displaystyle A} such that
⌊ A 3 n ⌋ {\displaystyle \lfloor A^{3^{n}}\rfloor }
is a prime number for all positive natural numbers n {\displaystyle n} (where ⌊ − ⌋ {\displaystyle \lfloor -\rfloor } denotes the floor function). This constant is named after William Harold Mills who in 1947 proved the existence of A {\displaystyle 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 … {\displaystyle 1.3063778838630806904686144926\dots }
(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 , {\displaystyle 2,11,1361,2521008887,16022236204009818131831320183,}
4113101149215104800030529537915953170486139623539759933135949994882770404074832568499 , … {\displaystyle 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,\ldots }
(sequence A051254 in the OEIS). If a i {\displaystyle a_{i}} denotes the i {\displaystyle i} th prime in this sequence, then a i {\displaystyle a_{i}} can be calculated as the smallest prime number larger than a i − 1 3 {\displaystyle a_{i-1}^{3}} . In order to ensure that rounding A 3 n {\displaystyle A^{3^{n}}} for all n ≥ 1 {\displaystyle n\geq 1} produces this sequence of primes, it must be the case that a i < ( a i − 1 + 1 ) 3 {\displaystyle 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 {\displaystyle 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 {\displaystyle a_{1}=2} . For all a i > e e 32.537 {\displaystyle a_{i}>e^{e^{32.537}}} , there is at least one prime between a i 3 {\displaystyle a_{i}^{3}} and ( a i + 1 ) 3 {\displaystyle (a_{i}+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
( ( ( ( ( ( ( ( ( 2 3 + 3 ) 3 + 30 ) 3 + 6 ) 3 + 80 ) 3 + 12 ) 3 + 450 ) 3 + 894 ) 3 + 3636 ) 3 + 70756 ) 3 + 97220 {\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. As of 2024, the largest known Mills probable prime (under the Riemann hypothesis) is
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