The Most Likely Common Difference of Arithmetic Progressions Among Primes

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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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. (2021). https://doi.org/10.1007/s40304-020-00218-3

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Keywords

  • Common difference
  • Arithmetic progression
  • Hardy–Littlewood Conjecture
  • Differences among primes
  • Singular series

Mathematics Subject Classification

  • 11N05
  • 11N13