2010/10/23 by Chris K. Caldwell, Caldwell, Chris K., Yuanyou Furui Cheng +1 · 1 citation
Mathematics · #11A41 #11Y11 #11Y60 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A41 #msc:11Y11 #msc:11Y60
paper · pdf · doi:10.48550/arxiv.1010.4883
9 pages, published on Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1
arxiv created 2013/01/25 · arxiv updated 2013/01/28
In 1947 Mills proved that there exists a constant A such that \lfloor A3n \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.