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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Dense admissible sequences


Authors: David A. Clark and Norman C. Jarvis
Journal: Math. Comp. 70 (2001), 1713-1718
MSC (2000): Primary 11B83, 11N13
Published electronically: March 22, 2001
MathSciNet review: 1836929
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Abstract:

A sequence of integers in an interval of length $x$ is called admissible if for each prime there is a residue class modulo the prime which contains no elements of the sequence. The maximum number of elements in an admissible sequence in an interval of length $x$ is denoted by $\varrho ^{*}(x)$. Hensley and Richards showed that $\varrho ^{*}(x)>\pi (x)$ for large enough $x$. We increase the known bounds on the set of $x$ satisfying $\varrho ^{*}(x)\le \pi (x)$ and find smaller values of $x$ such that $\varrho ^{*}(x)>\pi (x)$. We also find values of $x$ satisfying $\varrho ^{*}(x)>2\pi (x/2)$. This shows the incompatibility of the conjecture $\pi (x+y)-\pi (y)\le 2\pi (x/2)$ with the prime $k$-tuples conjecture.


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Additional Information

David A. Clark
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah, 84602
Email: clark@math.byu.edu

Norman C. Jarvis
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah, 84602
Email: jarvisn@math.byu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01348-5
PII: S 0025-5718(01)01348-5
Received by editor(s): August 5, 1996
Received by editor(s) in revised form: April 18, 1997
Published electronically: March 22, 2001
Additional Notes: After this paper was submitted, the authors learned that Dan Gordon and Gene Rodemich have extended the calculation of $𝜌^{*}(n)$ to $n=1600$.
Article copyright: © Copyright 2001 American Mathematical Society