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  • Bulletin of the Australian Mathematical Society, Volume 78, Issue 3
  • December 2008, pp. 431-436

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

  • XUE-GONG SUN (a1) (a2) and JIN-HUI FANG (a1)
  • DOI: http://dx.doi.org/10.1017/S0004972708000804
  • Published online: 01 December 2008
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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]Y. G. Chen , ‘On integers of the form 2n±p1α1prαr’, Proc. Amer. Math. Soc. 128 (2000), 16131616.

[2]Y. G. Chen , ‘On integers of the form k2n+1’, Proc. Amer. Math. Soc. 129 (2001), 355361.

[3]Y. G. Chen , ‘On integers of the forms k−2n and k2n+1’, J. Number Theory 89 (2001), 121125.

[4]Y. G. Chen , ‘On integers of the forms kr−2n and kr2n+1’, J. Number Theory 98 (2003), 310319.

[5]Y. G. Chen , ‘On integers of the forms k±2n and k2n±1’, J. Number Theory 125 (2007), 1425.

[6]Y. G. Chen , ‘Five consecutive positive odd numbers, none of which can be expressed as a sum of two prime powers’, Math. Comp. 74 (2005), 10251031.

[7]Y. G. Chen and X. G. Sun , ‘On Romanoff’s constant’, J. Number Theory 106 (2004), 275284.

[8]P. Erdős and A. M. Odlyzko , ‘On the density of odd integers of the form (p−1)2n and related questions’, J. Number Theory 11 (1979), 257263.

[9]R. K. Guy , Unsolved Problems in Number Theory, 3rd edn (Springer, New York, 2004).

[10]F. Luca and P. Stǎnicǎ , ‘Fibonacci numbers that are not sums of two prime powers’, Proc. Amer. Math. Soc. 133 (2005), 18871890.

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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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