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Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1 |
Yuanyou Cheng
Durham, North Carolina
USA
Abstract:
In 1947 Mills proved that there exists a constant such that
is a prime for every positive integer
.
Determining
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
does begin with
. We calculate this
value to
decimal places by determining the associated primes
to over
digits and probable primes (PRPs) to over
digits. We also apply the Cramér-Granville Conjecture to Honaker's
problem in a related context.
(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.