Rocky Mountain Journal of Mathematics

On the $\mod p^7$ determination of ${2p-1\choose p-1}$

Romeo Meštrović

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Abstract

In this paper we prove that for any prime $p\ge 11$, $$ {2p-1\choose p-1}\equiv 1 -2p \sum_{k=1}^{p-1}\frac{1}{k} +4p^2\sum_{1\le i\lt j\le p-1}\frac{1}{ij}\pmod{p^7} $$ holds. This is a generalization of the famous Wolstenholme's theorem which asserts that ${2p-1\choose p-1} \equiv 1 \ \pmod{\,p^3}$ for all primes $p\ge 5$. Our proof is elementary, and it does not use a standard technique involving the classic formula for power sums in terms of the Bernoulli numbers. Notice that the above congruence reduced modulo $p^6$, $p^5$ and $p^4$ yields related congruences obtained by Tauraso, Zhao and Glaisher, respectively.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 2 (2014), 633-648.

Dates
First available in Project Euclid: 4 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1407154917

Digital Object Identifier
doi:10.1216/RMJ-2014-44-2-633

Mathematical Reviews number (MathSciNet)
MR3240517

Zentralblatt MATH identifier
1372.11020

Subjects
Primary: 11B75: Other combinatorial number theory
Secondary: 11A07: Congruences; primitive roots; residue systems 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30] 11B68: Bernoulli and Euler numbers and polynomials 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx]

Keywords
Congruence prime power Wolstenholme's theorem Wolstenholme prime Bernoulli numbers

Citation

Meštrović, Romeo. On the $\mod p^7$ determination of ${2p-1\choose p-1}$. Rocky Mountain J. Math. 44 (2014), no. 2, 633--648. doi:10.1216/RMJ-2014-44-2-633. https://projecteuclid.org/euclid.rmjm/1407154917


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