We give an explicit form of Inghams Theorem on primes in the short intervals, and show that there is at least one prime between every two consecutive cubes $xsp{3}$ and $(x+1)sp{3}$ if $loglog xge 15$.
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 b
ooks 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 Cramer-Granville Conjecture to Honakers problem in a related context.
