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The Most Likely Common Difference of Arithmetic Progressions Among Primes

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Abstract

Let \(d^*_k(x)\) be the most likely common differences of arithmetic progressions of length \(k+1\) among primes \(\le x\). Based on the truth of Hardy–Littlewood Conjecture, we obtain that \(\lim \limits _{x\rightarrow +\infty }d^*_k(x)=+\infty \) uniformly in k, and every prime divides all sufficiently large most likely common differences.

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Acknowledgements

We would like to express our heartfelt thanks to the anonymous referees for their careful reading and helpful suggestion.

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, People’s Republic of China

    Xiaosheng Wu & Pengzhen Yang

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  1. Xiaosheng Wu
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  2. Pengzhen Yang
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Correspondence to Xiaosheng Wu.

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This work is supported by the National Natural Science Foundation of China (Grant No. 11871187).

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Wu, X., Yang, P. The Most Likely Common Difference of Arithmetic Progressions Among Primes. Commun. Math. Stat. 9, 315–329 (2021). https://doi.org/10.1007/s40304-020-00218-3

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  • DOI: https://doi.org/10.1007/s40304-020-00218-3

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