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

  • XUE-GONG SUN (a1) (a2) and JIN-HUI FANG (a1)
Abstract
Abstract

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)2n (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)2n have an asymptotic density of zero.

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      ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2n IN ARITHMETIC PROGRESSIONS
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References
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[1]Chen Y. G., ‘On integers of the form 2n±p 1α 1p rα r’, Proc. Amer. Math. Soc. 128 (2000), 16131616.
[2]Chen Y. G., ‘On integers of the form k2n+1’, Proc. Amer. Math. Soc. 129 (2001), 355361.
[3]Chen Y. G., ‘On integers of the forms k−2n and k2n+1’, J. Number Theory 89 (2001), 121125.
[4]Chen Y. G., ‘On integers of the forms k r−2n and k r2n+1’, J. Number Theory 98 (2003), 310319.
[5]Chen Y. G., ‘On integers of the forms k±2n and k2n±1’, J. Number Theory 125 (2007), 1425.
[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), 10251031.
[7]Chen Y. G. and Sun X. G., ‘On Romanoff’s constant’, J. Number Theory 106 (2004), 275284.
[8]Erdős P. and Odlyzko A. M., ‘On the density of odd integers of the form (p−1)2n and related questions’, J. Number Theory 11 (1979), 257263.
[9]Guy R. K., Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).
[10]Luca F. and Stǎnicǎ P., ‘Fibonacci numbers that are not sums of two prime powers’, Proc. Amer. Math. Soc. 133 (2005), 18871890.
[11]Sierpiński W., ‘Sur un problème concernant les nombres k2n+1’, Elem. Math. 15 (1960), 7374; Corrigendum 17 (1962), 85.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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