# Mills' theorem

In the late forties Mills proved [Mills47] that:

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

Here [ ] is the floor function.  This startlingly simple characterization of a sequence of primes is called Mills' Theorem.

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.

The difficulty in finding a value for A is that we need to construct a sequence of primes (Mills' primes) p1, p2, ...; for which pi-1 is between pi3 and (pi+1)3.  The constant A is then the limit of pn3-n as n approaches infinity.

So we need that there are primes between x3 and (x+1)3 for x > M for some bound M.  The best know lower bound may be 106000000000000000000 [Cheng2003a], but that is far far too large to allow actual calculations.  So for now we must ask "what is a likely value of A?"

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.  Folks usually assume our sequence of primes begin:

2,
11,
1361,
2521008887,
16022236204009818131831320183,
4113101149215104800030529537915953170486139623539759933135949994882770404074832568499,
695838043769627416085392765735385928648359...(254 digits)...257390268487534179757699110378097045955949,
336918228195740742277307753365919464724735980446...(762 digits)...405013138097469593692676561694614253113386536243,
...

(even though we can not prove this sequence continues without the Riemann Hypothesis).  This sequence is formed by choosing the minimal prime at each step, and yields the smallest possible value for Mills' constant:

1.30637788386308069046861449260260571291678...

Though amusing, this type of formula is useless for determining primes because we need to know the primes before we find A, and the subsequence of primes represented by Mills' theorem is so small.

Related pages (outside of this work)

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)
Ramare2002
O. Ramaré, "Éatu des lieux," (2002) available from http://math.univ-lille1.fr/~ramare/Maths/contenu.html..
Wright1954
Wright, E. M., "A class of representing functions," J. London Math. Soc., 29 (1954) 63--71.  MR 15,288d