TOPICS
Search

Mills' Prime


Mills' constant can be defined as the least theta such that

 f_n=|_theta^(3^n)_|

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

The first few f_n for n=1, 2, ... are 2, 11, 1361, 2521008887, ... (OEIS A051254). They can be represented more compactly through b_n as f_1=2 and

 f_(n+1)=f_n^3+b_n.

Caldwell and Cheng (2005) calculated the first 10 Mills primes. 13 are known as of Jul. 2013, with the firth few b_n for n=1, 2, ... being 3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, ... (OEIS A108739). b_(13) is not known, but it is known that b_(13)>221 (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).


See also

Elliptic Curve Primality Proving, Mills' Constant

Explore with Wolfram|Alpha

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. http://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."

Referenced on Wolfram|Alpha

Mills' Prime

Cite this as:

Weisstein, Eric W. "Mills' Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MillsPrime.html

Subject classifications