The Most Likely Common Difference of Arithmetic Progressions Among Primes


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Erdös, P., Straus, E.G.: Remarks on the differences between consecutive primes. Elem. Math. 35(5), 115–118 (1980)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Feng, S., Wu, X.: The \(k\)-tuple jumping champions among consecutive primes. Acta. Arith. 156(4), 325–339 (2012)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Funkhouser, S., Goldston, D.A., Sengupta, D., Sengupta, J.: Prime difference champions. arXiv:1612.02938

  4. 4.

    Goldston, D.A., Pintz, J., Yildirim, C.Y.: Primes in tuples. I. Ann. Math. 170, 819–862 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Goldston, D.A., Ledoan, A.H.: Jumping champions and gaps between consecutive primes. Int. J. Number Theory 7(6), 1–9 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Goldston, D.A., Ledoan, A.H.: The jumping champions conjecture. Mathematika 61(3), 719–740 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Granville, A., Kane, D.M., Koukoulopoulos, D., Lemke Oliver, R.J.: Best Possible Dersities of Dickson \(m\)-Tuples, as a Consequence of Zhang-Maynard-Tao. Analytic Number Theory, pp. 133–144. Springer, Cham (2015)

    Google Scholar 

  8. 8.

    Green, B.J., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Ann. Math. 167(6), 481–547 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes. Acta. Math. 44(1), 1–70 (1923). Reprinted as pp. 561–630 in Collected Papers of G. H. Hardy, Vol. I (Editor by a committee appointed the London Mathematical Society), Clarendon Press, Oxford (1966)

  10. 10.

    Huang, W., Shao, S., Ye, X.: Nil Bohr-sets and almost automorphy of higher order. Mem. Am. Math. Soc. 241(1143), 83 (2016)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Huang, W., Wu, X.: On the set of the difference of primes. Proc. Am. Math. Soc. 145(9), 3787–3793 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Kronecker, L.: Vorlesungern über Zahlenthorie, I., p. 68, Teubner, Leipzig (1901)

  13. 13.

    Maynard, J.: Small gaps between primes. Ann. Math. (2) 181(1), 383–413 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Nelson, H.: Problem 654: Consecutive primes. J. Recr. Math. 10, 212 (1977–78)

  15. 15.

    Odlyzko, A., Rubinstein, M., Wolf, M.: Jumping champions. Exp. Math. 8(2), 107–118 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Pintz, J.: Polignac Numbers, Conjectures of Erdös on Gaps Between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture. From Arithmetic to Zeta-Functions, pp. 367–384. Springer, Cham (2016)

    Google Scholar 

  17. 17.

    Polymath, D.H.J.: Variants of the Selberg sieve, and bounded intervals containing many primes. Res. Math. Sci. 1, 83 (2014), Art. 12

  18. 18.

    Shao, X.: Narrow arithmetic progressions in the primes. Int. Math. Res. Not. IMRN 2, 391–428 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Tao, T., Ziegler, T.: Narrow Progressions in the Primes. Analytic Number Theory, pp. 357–379. Springer, Berlin (2015)

    Google Scholar 

  20. 20.

    Wu, L., Wu, X.: The \(k\)-tuple prime difference champions. J. Number Theory 196, 223–243 (2019)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Zhang, Y.: Bounded gaps between primes. Ann. Math. 179, 1121–1174 (2014)

    MathSciNet  MATH  Article  Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Xiaosheng Wu.

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wu, X., Yang, P. The Most Likely Common Difference of Arithmetic Progressions Among Primes. Commun. Math. Stat. (2021).

Download citation


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

Mathematics Subject Classification

  • 11N05
  • 11N13