New results on Wolstenholmes theorem and its applications

Document Type : Review Paper

Authors

1 Department of Computer Engineering, University of Torbat e Jam, , Iran

2 Department of Mathematics, Ferdowsi University of Mashhad, Iran.

Abstract

In 1862, Wolstenholme proved that for primes $p\geq 5$, $\binom{2p-1}{p-1}\equiv 1\pmod{p^3}$. This theorem is equivalent to the divisibility of the coefficients of the harmonic series \[1+\frac{1}{2}+\cdots+\frac{1}{p-1}\] by $p^2$. The far-reaching implications of Wolstenholme's theorem ignited the curiosity of mathematicians in the 19th century, leading to a surge of investigations into the divisibility properties of coefficients in rational fractions and binomial coefficients involving powers of primes. The emergence of Bernoulli numbers in this theorem and their intricate connection to binomial coefficients firmly established the roots of this mathematical domain in analytic number theory. In this paper, we embark on a journey to explore Wolstenholme's theorem for higher powers of primes and delve into the intricacies of these diverse proofs. The converse of Wolstenholme's theorem, first proposed by Jones, asserts that a natural number n satisfying the congruence $\binom{2n-1}{n-1}\equiv 1\pmod{p^3}$ must be prime. We conclude our exploration by examining the conditions of the converse of Wolstenholme's theorem and presenting several intriguing problems that beckon further investigation.

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Main Subjects

References

[1] T. M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.
[2] C. Babbage, Demonstration of a theorem relating to prime numbers, Edinburgh. Phil. J., 1 (1819) 46–49.
[3] M. Bayat, A generalization of Wolstenholme’s theorem, Amer. Math. Monthly, 104 no. 6 (1997) 557–560.
[4] V. Brun, J. O. Stubban, J. E. Fjeldstad, R. Tambs Lyche, K. E. Aubert, W. Ljunggren and E. Jacobsthal, On the divisibility of the difference between two binomial coefficients, Den 11te Skandinaviske Matematikerkongress, Trondheim, (1949), Johan Grundt Tanums Forlag, Oslo, (1952) 42–54.
[5] L. Carlitz, A Note on Stirling Numbers of the First Kind, Math. Mag., 37 no. 5 (1964) 318–321.
[6] L. Comtet, Advanced Combinatorics: The art of finite and infinite expansions, Springer Science & Business Media, 2012.
[7] J. W. L. Glaisher, Congruences relating to the sums of products of the first n numbers and to other sums of products, Quart. J. Math., 31 (1900) 1–35.
[8] J. W. L. Glaisher, On the residues of the sums of products of the first p − 1 numbers, and their powers, to modulus p2 or p3, Quart. J. Math., 31 (1900) 321–353.
[9] R. K. Guy, Unsolved problems in number theory, Problem Books in Mathematics. Unsolved Problems in Intuitive Mathematics, 1, Springer-Verlag, New York-Berlin, 1981.
[10] C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory, 128 no. 3 (2008) 475–499.
[11] K. Ireland and M. Rosen, classical introduction to modern number theory, Revised edition of Elements of number theory, Graduate Texts in Mathematics, 84, Springer-Verlag, New York-Berlin, 1982.
[12] J. P. Jone, Private correspondence, January, 1994.
[13] R. Meˇstrovi´c, Congruences forWolstenholme primes, Czechoslovak Math. J., 65(140) no. 1 (2015) 237–253.
[14] R. J. McIntosh, On the converse of Wolstenholme’s theorem, Acta Arith., 71 no. 4 (1995) 381–389.
[15] M. Mirzavaziri Counting the Countable, Sokhangostar Publication, Tehran, Iran, 2008. [InPersion]
[16] M. P. Saikia, Conjectures on congruences of binomial coefficients modulo higher powers of a prime number, J. Assam Acad. Math., 13 (2023) 5–7.
[17] R. Tauraso, More congruences for central binomial coefficients, J. Number Theory, 130 no. 12 (2010) 2639–2649.
[18] J. Wolstenholme, On certain properties of prime numbers, Quart. J. Pure Appl.Math., 5 (1862) 35–39.
[19] D. Yaqubi and M. Mirzavaziri, Extending Wolstenholm’s theorem: A new perspective, (2024). https://www.researchgate.net/publication/386148476.
[20] D. Yaqubi and M. Mirzavaziri, Some divisibility properties of binomial coefficients, J. Number Theory, 183 (2018) 428–441.
[21] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 no. 1 (2008) 73–106.
[22] Z. Zhi-Hong, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 no. 1-3 (2000) 193–223.
[23] V. Trevisan, W. Kenneth, Testing the converse of Wolstenholme’s theorem. 16th School of Algebra, Part II (Portuguese) (Bras´ılia, 2000), Mat. Contemp., 21 (2001) 275–286.
Mathematics and Society
Volume 10, Issue 4 - Serial Number 4
December 2025
Pages 105-131

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History

  • Receive Date: 10 June 2024
  • Revise Date: 17 January 2025
  • Accept Date: 27 January 2025
  • Publish Date: 20 February 2026

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