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On the density of integers of the form 2k + p in arithmetic progressions

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Abstract

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

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Authors and Affiliations

  1. Department of Mathematics and Science, Huai Hai Institute of Technology, Lian Yun Gang, 222005, P. R. China

    Xue Gong Sun

  2. Department of Mathematics, Nanjing Normal University, Nanjing, 210097, P. R. China

    Xue Gong Sun

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  1. Xue Gong Sun
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Correspondence to Xue Gong Sun.

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Supported by National Natural Science Foundation of China (Grant Nos. 10771103 and 10801075)

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Sun, X.G. On the density of integers of the form 2k + p in arithmetic progressions. Acta. Math. Sin.-English Ser. 26, 155–160 (2010). https://doi.org/10.1007/s10114-010-8013-y

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  • DOI: https://doi.org/10.1007/s10114-010-8013-y

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