Mills' Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Mills (1947) proved the existence of a constant $\theta=1.3064\dots$ such that

$$\left\lfloor{\theta^{3^n}}\right\rfloor$$ (1)

is Prime for all $n$ $\geq 1$, where $\left\lfloor{x}\right\rfloor$ is the Floor Function. It is not, however, known if $\theta$ is Irrational. Mills' proof was based on the following theorem by Hoheisel (1930) and Ingham (1937). Let $p_n$ be the $n$th Prime, then there exists a constant $K$ such that

$$p_{n+1}-p_n<K {p_n}^{5/8}$$ (2)

for all $n$. This has more recently been strengthened to

$$p_{n+1}-p_n<K {p_n}^{1051/1920}$$ (3)

(Mozzochi 1986). If the Riemann Hypothesis is true, then Cramér (1937) showed that

$$p_{n+1}-p_n={\mathcal O}(\ln p_n\sqrt{p_n}\,)$$ (4)

(Finch).

Hardy and Wright (1979) point out that, despite the beauty of such Formulas, they do not have any practical consequences. In fact, unless the exact value of $\theta$ is known, the Primes themselves must be known in advance to determine $\theta$. A generalization of Mills' theorem to an arbitrary sequence of Positive Integers is given as an exercise by Ellison and Ellison (1985). Consequently, infinitely many values for $\theta$ other than the number $1.3064\ldots$ are possible.

References

Caldwell, C. ``Mills' Theorem--A Generalization.'' http://www.utm.edu/research/primes/notes/proofs/A3n.html.

Ellison, W. and Ellison, F. Prime Numbers. New York: Wiley, pp. 31-32, 1985.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/mills/mills.html

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Mills, W. H. ``A Prime-Representing Function.'' Bull. Amer. Math. Soc. 53, 604, 1947.

Mozzochi, C. J. ``On the Difference Between Consecutive Primes.'' J. Number Th. 24, 181-187, 1986.

Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 135 and 191-193, 1989.

Ribenboim, P. The Little Book of Big Primes. New York: Springer-Verlag, pp. 109-110, 1991.