Mills' Prime

Mills' constant can be defined as the least  such that

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).

REFERENCES:

Caldwell, C. K. and Cheng, Y. "Determining Mills' Constant and a Note on Honaker's Problem." J. Integer Sequences 8, Article 05.4.1, 1-9, 2005. https://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html.

Sloane, N. J. A. Sequences A051254, A108739, and A224845 in "The On-Line Encyclopedia of Integer Sequences."