Skip to main content
SpringerLink
Log in

Wolstenholme and Vandiver primes

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

A prime p is a Wolstenholme prime if \(\left( {\begin{array}{c}2p\\ p\end{array}}\right) \equiv 2\) mod \(p^4\), or, equivalently, if p divides the numerator of the Bernoulli number \(B_{p-3}\); a Vandiver prime p is one that divides the Euler number \(E_{p-3}\). Only two Wolstenholme primes and eight Vandiver primes are known. We increase the search range in the first case by a factor of ten, and show that no additional Wolstenholme primes exist up to \({10^{11}}\), and in the second case by a factor of twenty, proving that no additional Vandiver primes occur up to this same bound. To facilitate this, we develop a number of new congruences for Bernoulli and Euler numbers mod p that are favorable for computation, and we implement some highly parallel searches using GPUs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. Some authors instead use the series for the hyperbolic secant, \({{\,\mathrm{sech}\,}}z = ({2e^z})/({\mathrm{{e}}^{2z}+1})\), to define the Euler numbers; this alters the sign of the terms with index congruent to 2 mod 4. Results cited from the literature that employ that formulation are suitably translated here to comport with the definition we employ.

  2. We remark that some structure does appear in a similar problem. The Ankeny–Artin–Chowla conjecture asserts that \(p\not \mid B_{(p-1)/2}\) for every prime \(p\equiv 1\) mod 4. While this problem is open in general (and has been verified computationally for \(p<2\cdot 10^{11}\) [29]), this conjecture is known to hold for primes of the form \(p=n^2+1\) or \(n^2+4\) [2].

References

  1. Aebi, C., Cairns, G.: Wolstenholme again. Elem. Math. 70(3), 125–130 (2015)

    Article  MathSciNet  Google Scholar 

  2. Agoh, T.: Congruences related to the Ankeny-Artin-Chowla conjecture, Integers, 16, paper no. A12, 30 pp (2016)

  3. Buhler, J., Crandall, R., Ernvall, R., Metsänkylä, T.: Irregular primes and cyclotomic invariants to four million. Math. Comp. 61(203), 151–153 (1993)

    Article  MathSciNet  Google Scholar 

  4. Carlitz, L.: Note on irregular primes. Proc. Amer. Math. Soc. 5, 329–331 (1954)

    Article  MathSciNet  Google Scholar 

  5. Cosgrave, J.B., Dilcher, K.: On a congruence of Emma Lehmer related to Euler numbers. Acta Arith. 161(1), 47–67 (2013)

    Article  MathSciNet  Google Scholar 

  6. Edwards, H. M.: Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, Grad. Texts in Math., 50, Springer, New York, xvi+410 (1996)

  7. Ernvall, R.: Corrigenda: “Cyclotomic invariants and \(E\)-regular primes”, Math. Comp., 33(145), 433 (1979)

  8. Ernvall, R., Metsänkylä, T.: Cyclotomic invariants and \(E\)-irregular primes. Math. Comp. 32(142), 617–629 (1978)

    MathSciNet  MATH  Google Scholar 

  9. Gardiner, A.: Four problems on prime power divisibility. Amer. Math. Monthly 95(10), 926–931 (1988)

    Article  MathSciNet  Google Scholar 

  10. Glaisher, J.W.L.: Congruences relating to the sums of products of the first \(n\) numbers and to other sums of products. Quart. J. Pure Appl. Math. 31, 1–35 (1900)

    MATH  Google Scholar 

  11. Glaisher, J.W.L.: On the residues of the sums of products of the first \(p-1\) numbers, and their powers, to modulus \(p^2\) or \(p^3\). Quart. J. Pure Appl. Math. 31, 321–353 (1900)

    MATH  Google Scholar 

  12. Glaisher, J.W.L.: On the residues of the sums of the inverse powers of numbers in arithmetic progression. Quart. J. Pure Appl. Math. 32, 271–305 (1901)

    MATH  Google Scholar 

  13. Gut, M.: Euler’sche Zahlen und Klassenanzahl des Körpers der 4\(\ell \)-ten Einheitswurzeln. Comment. Math. Helv. 25, 43–63 (1951)

    Article  MathSciNet  Google Scholar 

  14. Hart, W., Harvey, D., Ong, W.: Irregular primes to two billion. Math. Comp. 86(308), 3031–3049 (2017)

    Article  MathSciNet  Google Scholar 

  15. Johnson, W.: Irregular primes and cyclotomic invariants. Math. Comp. 29, 113–120 (1975)

    Article  MathSciNet  Google Scholar 

  16. Lehmer, E.: On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson. Ann. of Math. (2) 39(2), 350–360 (1938)

  17. Luca, F., Pizarro-Madariaga, A., Pomerance, C.: On the counting function of irregular primes, Indag. Math. (N.S.), 26(1), 147–161 (2015)

  18. McIntosh, R.J.: On the converse of Wolstenholme’s theorem. Acta Arith. 71(4), 381–389 (1995)

    Article  MathSciNet  Google Scholar 

  19. McIntosh, R. J.: Congruences involving Euler numbers and power sums, Fibonacci Quart., 58(4), 328–333 (2020)

  20. McIntosh, R.J., Roettger, E.L.: A search for Fibonacci-Wieferich and Wolstenholme primes. Math. Comp. 76(260), 2087–2094 (2007)

    Article  MathSciNet  Google Scholar 

  21. Meštrović, R.: Wolstenholme’s theorem: its generalizations and extensions in the last hundred and fifty years (1862–2012), arXiv:1111.3057v2, (25 Dec. 2011), 31 pp

  22. Meštrović, R.: A search for primes \(p\) such that the Euler number \(E_{p-3}\) is divisible by \(p\). Math. Comp. 83(290), 2967–2976 (2014)

    Article  MathSciNet  Google Scholar 

  23. Selfridge, J. L., Pollack, B. W.: Fermat’s last theorem is true for any exponent up to 25,000, Notices Amer. Math. Soc., 11, no. 1, (1964), 97, Abstract no. 608-138, Annual Meeting in Miami and Coral Gables, FL, Jan. 23-26, 1964

  24. Stafford, E.T., Vandiver, H.S.: Determination of some properly irregular cyclotomic fields. Proc. Natl. Acad. Sci. USA 16(2), 139–150 (1930)

    Article  Google Scholar 

  25. Sun, Z.-H.: Congruences involving Bernoulli and Euler numbers. J. Number Theory 128(2), 280–312 (2008)

  26. Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54(12), 2509–2535 (2011)

    Article  MathSciNet  Google Scholar 

  27. Tanner, J.W., Wagstaff Jr., S.S.: New congruences for the Bernoulli numbers. Math. Comp. 48(177), 341–350 (1987)

    Article  MathSciNet  Google Scholar 

  28. Uspensky, J. V., Heaslet, M. A.: Elementary Number Theory, McGraw-Hill, New York, x+484 (1939)

  29. Van Der Poorten, A. J., te Riele, H. J. J., Williams, H. C.: Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than \(100\,000\,000\,000\), Math. Comp., 70(235), 1311–1328 (2001) Corrigenda and addition, ibid. 72 (241), 521–523 (2003)

  30. Vandiver, H.S.: On Bernoulli’s numbers and Fermat’s last theorem. Duke Math. J. 3(4), 569–584 (1937)

    Article  MathSciNet  Google Scholar 

  31. Vandiver, H.S.: Note on Euler number criteria for the first case of Fermat’s last theorem. Amer. J. Math. 62, 79–82 (1940)

    Article  MathSciNet  Google Scholar 

  32. Wagstaff Jr., S.S.: The irregular primes to \(125000\). Math. Comp. 32(142), 583–591 (1978)

    Article  MathSciNet  Google Scholar 

  33. Walisch, K.: Primesieve, ver. 7.6 (Jan. 7, 2021), https://github.com/kimwalisch/primesieve

  34. Washington, L. C.: Introduction to Cyclotomic Fields, Grad. Texts in Math., 83, 2nd ed. Springer, New York, xiv+487 (1997)

  35. Wolstenholme, J.: On certain properties of prime numbers. Quart. J. Pure Appl. Math. 5, 35–39 (1862)

    Google Scholar 

Download references

Acknowledgements

We thank Karl Dilcher, Lars Hesselholt, and Richard McIntosh for helpful correspondence. We also thank NCI Australia and UNSW Canberra for computational resources. This research was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government.

Author information

Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, UK

    Andrew R. Booker

  2. School of Science, The University of New South Wales, Canberra, Australia

    Shehzad Hathi & Timothy S. Trudgian

  3. Center for Communications Research, Princeton, NJ, USA

    Michael J. Mossinghoff

Authors
  1. Andrew R. Booker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shehzad Hathi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael J. Mossinghoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Timothy S. Trudgian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Timothy S. Trudgian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by Australian Research Council Future Fellowship FT160100094.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Booker, A.R., Hathi, S., Mossinghoff, M.J. et al. Wolstenholme and Vandiver primes. Ramanujan J 58, 913–941 (2022). https://doi.org/10.1007/s11139-021-00438-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-021-00438-3

Keywords

Mathematics Subject Classification

Search

Navigation