Mills' Constant

Mills' theorem states that there exists a real constant such that is prime for all positive integers (Mills 1947). While for each value of , there are uncountably many possible values of such that is prime for all positive integers (Caldwell and Cheng 2005), it is possible to define Mills' constant as the least such that

is prime for all positive integers , giving a value of

(OEIS A051021).

is therefore given by the next prime after , and the values of are known as Mills' primes (Caldwell and Cheng 2005).

Caldwell and Cheng (2005) computed more than 6850 digits of assuming the truth of the Riemann hypothesis. Proof of primality of the 13 Mills prime in Jul. 2013 means that approximately digits are now known.

It is not known if is irrational.