Mills' prime

In the late forties Mills proved [Mills47] that:

Mills' Theorem: there is a real number A for which [] is always a prime (n = 1,2,3,...). 

The key to finding a value for A is that we need to construct a sequence of primes (Mills' primes) p₁, p₂, ...; for which p_i-1 is between p_i³ and (p_i+1)³.  The constant A is then the limit of p_n^3-n as n approaches infinity.

Mills' one page article contained no numerics, it only proved the existence of A.  Others [Wright1954] later proved that there are uncountably many choices for A, but again gave no value for A. In fact, we can only prove there are primes between consecutive cubes if we are looking at numbers beyond 10^{6000000000000000000} [Cheng2003a].

If we assume the Riemann Hypothesis, then it is easy to show there are primes between consecutive cubes of integers [CC2005] greater than one; and then we can calculate example to our hearts content.  Following tradition (and definitley not following Mills) we begin our sequence of primes as follows:

b₁ = 2,
b₂ = 11,
b₃ = 1361,
b₄ = 2521008887,
b₅ = 160222362 0400981813 1831320183,
b₆ = 41131 0114921510 4800030529 5379159531 7048613962 3539759933 1359499948 8277040407 4832568499

These are called the Mills Primes.  This sequence is formed by choosing the minimal prime at each step, and yields the smallest possible value for Mills' constant:

1.30637788386308069046861449260260571291678...

A few more Mills primes are known.  To make them easier to present, let b_n+1 = b_n³+a_n.  The sequence a_n begins:

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220 (b_n prp), 66768 (b_n prp)

The primality of b₇, b₈, and b₉ (2285 digits) were proved byBouk de Water in 2000; and Morain proved b₉ (6854 digits) prime in 2005. Carmody found the two PRP's (20562 and 61684 digits) in 2004.

See Also: MillsTheorem, MillsConstant

Related pages (outside of this work)

• Mills' Theorem - a generalization (and proof)
• Six thousand digits of (the least) Mills' Constant

References:

CC2005

C. Caldwell and Y. Cheng, "Determining Mills' constant and a note on Honaker's problem," J. Integer Seq., 8:4 (2005) Article 05.4.1, 9 pp. (electronic).  Available from http://www.cs.uwaterloo.ca/journals/JIS/.  MR2165330 (Abstract available)

Cheng2003a

Cheng, Yuan-You Fu-Rui, "Explicit estimate on primes between consecutive cubes," Rocky Mountain J. Math., 40:1 (2010) 117--153.  (http://dx.doi.org/10.1216/RMJ-2010-40-1-117) MR 2607111

Mills47

W. H. Mills, "A prime-representing function," Bull. Amer. Math. Soc., 53 (1947) 604.  MR 8,567d (Annotation available)

Wright1954

Wright, E. M., "A class of representing functions," J. London Math. Soc., 29 (1954) 63--71.  MR 15,288d