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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
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

References

[1]Adleman, L. M., Pomerance, C. and Rumely, R. S., ‘On distinguishing prime numbers from composite numbers’, Ann. of Math. (1) 117 (1983), 173–206.CrossRefGoogle Scholar
[2]Bateman, P. T. and Diamond, H., Analytic Number Theory: An Introductory Course (World Scientific, Singapore, 2004).CrossRefGoogle Scholar
[3]Chen, Y. G., ‘On integers of the form k⋅2n+1’, Proc. Amer. Math. Soc. 129 (2000), 355–361.CrossRefGoogle Scholar
[4]Davenport, H., Multiplicative Number Theory, 3rd edn (Springer, New York, 2000).Google Scholar
[5]Erdős, P. and Odlyzko, A. M., ‘On the density of odd integers of the form (p−1)2−n and related questions’, J. Number Theory 11 (1979), 257–263.CrossRefGoogle Scholar
[6]Hardy, G. H. and Littlewood, J. E., ‘Some problems of “Partitio numerorum”; III: On the expression of a number as a sum of primes’, Acta Math. 44 (1923), 1–70.CrossRefGoogle Scholar
[7]Heath-Brown, D. R., ‘Artin’s conjecture for primitive roots’, Q. J. Math. Oxford Ser. (2) 37 (1986), 27–38.CrossRefGoogle Scholar
[8]Mirsky, L., ‘The number of representations of an integer as the sum of a prime and a k-free integer’, Amer. Math. Monthly 56 (1949), 17–19.CrossRefGoogle Scholar
[9]Nathanson, M. B., Additive Number Theory: The Classical Bases (Springer, New York, 1996).CrossRefGoogle Scholar
[10]Ribenboim, P., The New Book of Prime Number Records, 3rd edn (Springer, New York, 1996).CrossRefGoogle Scholar
[11]Rosser, J. B. and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
[12]Sierpiński, W., ‘Sur un problème concernant les nombres k⋅2n+1’, Elem. Math. 15 (1960), 73–74.Google Scholar
[13]Suryanarayana, D., ‘The number of k-free integers ≤x which are coprime to n’, Indian J. Math. 11 (1969), 131–139.Google Scholar
[14]Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory (Cambridge University Press, Cambridge, 1995).Google Scholar