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ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

Part of: Additive number theory; partitions Elementary number theory Sequences and sets

Published online by Cambridge University Press:  01 December 2008

XUE-GONG SUN  and
JIN-HUI FANG
XUE-GONG SUN
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China Department of Mathematics and Science, Huai Hai Institute of Technology, Lian Yun Gang 222005, Jiangsu, People’s Republic of China (email: [email protected])
JIN-HUI FANG
Affiliation:
Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
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Abstract

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Erdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Keywords

asymptotic densitycovering systemsarithmetic progressions

MSC classification

Secondary: 11A07: Congruences; primitive roots; residue systems 11B25: Arithmetic progressions 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 3 , December 2008 , pp. 431 - 436
Copyright
Copyright © 2009 Australian Mathematical Society

References

[1]Chen, Y. G., ‘On integers of the form 2n±p 1α 1⋯p rα r’, Proc. Amer. Math. Soc. 128 (2000), 1613–1616.CrossRefGoogle Scholar
[2]Chen, Y. G., ‘On integers of the form k2n+1’, Proc. Amer. Math. Soc. 129 (2001), 355–361.CrossRefGoogle Scholar
[3]Chen, Y. G., ‘On integers of the forms k−2n and k2n+1’, J. Number Theory 89 (2001), 121–125.CrossRefGoogle Scholar
[4]Chen, Y. G., ‘On integers of the forms k r−2n and k r2n+1’, J. Number Theory 98 (2003), 310–319.CrossRefGoogle Scholar
[5]Chen, Y. G., ‘On integers of the forms k±2n and k2n±1’, J. Number Theory 125 (2007), 14–25.CrossRefGoogle Scholar
[6]Chen, Y. G., ‘Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers’, Math. Comp. 74 (2005), 1025–1031.CrossRefGoogle Scholar
[7]Chen, Y. G. and Sun, X. G., ‘On Romanoff’s constant’, J. Number Theory 106 (2004), 275–284.CrossRefGoogle Scholar
[8]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
[9]Guy, R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).CrossRefGoogle Scholar
[10]Luca, F. and Stǎnicǎ, P., ‘Fibonacci numbers that are not sums of two prime powers’, Proc. Amer. Math. Soc. 133 (2005), 1887–1890.CrossRefGoogle Scholar
[11]Sierpiński, W., ‘Sur un problème concernant les nombres k2n+1’, Elem. Math. 15 (1960), 73–74; Corrigendum 17 (1962), 85.Google Scholar