The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins If ai denotes the ith prime in this sequence, then ai can be calculated as the smallest prime number larger than. In order to ensure that rounding, for n = 1, 2, 3, …, produces this sequence of primes, it must be the case that. 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. 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 >, there is at least one prime between and . 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 and has 20562 digits. , the largest known Mills probable prime is , which is 555,154 digits long.
Numerical calculation
By calculating the sequence of Mills primes, one can approximate Mills' constant as used this method to compute 6850 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. If it is rational, and if we can calculate its decimal expansion to the point where it repeats, this will allow us to generate infinitely many provable primes.
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 . Matomäki showed unconditionally the existence of a constant A such that is prime for all n. Additionally, Tóth proved that the floor function in the formula could be replaced with the ceiling function, so that there exists a constant such that is also prime-representing for . In the case, the value of the constant begins with 1.24055470525201424067... The first few primes generated are:
