is prime for all positive integers (Caldwell and Cheng 2005).

The first few
for , 2, ... are 2, 11, 1361, 2521008887,
... (OEIS A051254). They can be represented
more compactly through
as and

Caldwell and Cheng (2005) calculated the first 10 Mills primes. 13 are known as of Jul. 2013, with the firth few for , 2, ... being 3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220,
66768, 300840, ... (OEIS A108739). is not known, but it is known that (E. Weisstein, Aug. 13, 2013).

The integer lengths of the Mills' primes are 1, 2, 4, 10, 29, 85, 254, 762, 2285,
6854, 20562, 61684, 185052, ... (OEIS A224845).