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

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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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