Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1

Determining Mills' Constant and a Note on Honaker's Problem


Chris K. Caldwell
Department of Mathematics and Statistics
University of Tennessee at Martin
Martin, TN 38238
USA

Yuanyou Cheng
Durham, North Carolina
USA

Abstract: In 1947 Mills proved that there exists a constant $A$ such that $\lfloor A^{3^n} \rfloor$ is a prime for every positive integer $n$. Determining $A$ requires determining an effective Hoheisel type result on the primes in short intervals--though most books ignore this difficulty. Under the Riemann Hypothesis, we show that there exists at least one prime between every pair of consecutive cubes and determine (given RH) that the least possible value of Mills' constant $A$ does begin with $1.3063778838$. We calculate this value to $6850$ decimal places by determining the associated primes to over $6000$ digits and probable primes (PRPs) to over $60000$ digits. We also apply the Cramér-Granville Conjecture to Honaker's problem in a related context.

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(Concerned with sequences A051021 A051254 and A108739 .)

Received July 14 2005; revised version received August 15 2005. Published in Journal of Integer Sequences August 24 2005.

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