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FLAT PRIMES AND THIN PRIMES

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  26 April 2010

KEVIN A. BROUGHAN  and
QIZHI ZHOU
KEVIN A. BROUGHAN*
Affiliation:
University of Waikato, Hamilton, New Zealand (email: kab@waikato.ac.nz)
QIZHI ZHOU
Affiliation:
University of Waikato, Hamilton, New Zealand (email: qz49@waikato.ac.nz)
*
✉For correspondence; e-mail: kab@waikato.ac.nz
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Abstract

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A number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges.

Keywords

flat primethin primesieve

MSC classification

Secondary: 11A41: Primes 11B83: Special sequences and polynomials

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 282 - 292
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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