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AN EXTENSION OF SURY’S IDENTITY AND RELATED CONGRUENCES

Part of: Combinatorics Sequences and sets Elementary number theory

Published online by Cambridge University Press:  04 October 2011

ROMEO MEŠTROVIĆ
ROMEO MEŠTROVIĆ*
Affiliation:
Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro (email: [email protected])
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Abstract

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In this paper we give an extension of a curious combinatorial identity due to B. Sury. Our proof is very simple and elementary. As an application, we obtain two congruences for Fermat quotients modulo p3. Moreover, we prove an extension of a result by H. Pan that generalizes Carlitz’s congruence.

Keywords

Sury’s combinatorial identityharmonic numberFermat quotientCarlitz’s congruence

MSC classification

Secondary: 11A07: Congruences; primitive roots; residue systems 05A10: Factorials, binomial coefficients, combinatorial functions 11B65: Binomial coefficients; factorials; $q$-identities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 3 , June 2012 , pp. 482 - 496
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Cai, T. X. and Granville, A., ‘On the residues of binomial coefficients and their products modulo prime powers’, Acta Math. Sin. (Engl. Ser.) 18 (2002), 277–288.CrossRefGoogle Scholar
[2]Carlitz, L., ‘A theorem of Glaisher’, Canad. J. Math. 5 (1953), 306–316.CrossRefGoogle Scholar
[3]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’, Q. J. Math. 31 (1900), 321–353.Google Scholar
[4]Gould, H. W., Combinatorial Identities, a Standardized Set of Tables Listing 500 Binomial Coefficient Summations, 2nd edn (Published by the author, West Virginia University, Morgantown, 1972).Google Scholar
[5]Granville, A., ‘The square of the Fermat quotient’, Integers 4 (2004), Article A22.Google Scholar
[6]Morley, F., ‘Note on the congruence 24n≡(−1)n(2n)!/(n!)2, where 2n+1 is a prime’, Ann. of Math. (2) 9 (1895), 168–170.CrossRefGoogle Scholar
[7]Pan, H., ‘On a generalization of Carlitz’s congruence’, Int. J. Mod. Math. 4 (2009), 87–93.Google Scholar
[8]Staver, T. B., ‘Om summasjon av potenser av binomialkoeffisienten’, Normat 29 (1947), 97–103.Google Scholar
[9]Sun, Z. H., ‘Congruences concerning Bernoulli numbers and Bernoulli polynomials’, Discrete Appl. Math. 105 (2000), 193–223.CrossRefGoogle Scholar
[10]Sun, Z. H., ‘Congruences involving Bernoulli and Euler numbers’, J. Number Theory 128 (2008), 280–312.CrossRefGoogle Scholar
[11]Sun, Z. W., ‘Arithmetic theory of harmonic numbers’, Proc. Amer. Math. Soc. (2009), article in press, S 0002-9939(2011) 10925-0; preprint, arXiv: 0911.4433v6 [math.NT].Google Scholar
[12]Sun, Z. W., ‘Binomial coefficients, Catalan numbers and Lucas quotients’, Sci. China Math. 53 (2010), 2473–2488.CrossRefGoogle Scholar
[13]Sury, B., ‘Sum of the reciprocals of the binomial coefficients’, European J. Combin. 14 (1993), 351–353.CrossRefGoogle Scholar
[14]Sury, B., Wang, T. and Zhao, F. Z., ‘Identities involving reciprocals of binomial coefficients’, J. Integer Seq. 7 (2004), Article 04.2.8.Google Scholar
[15]Tauraso, R., ‘New harmonic number identities with applications’, Sém. Lothar. Combin. 63 (2010), Article B63g.Google Scholar
[16]Yang, J. H. and Zhao, F. Z., ‘Sums involving the inverses of binomial coefficients’, J. Integer Seq. 9 (2006), Article 06.4.2.Google Scholar
[17]Yang, J. H. and Zhao, F. Z., ‘Certain sums involving inverses of binomial coefficients and some integrals’, J. Integer Seq. 10 (2007), Article 07.8.7.Google Scholar
[18]Zhao, F. Z. and Wang, T., ‘Some results for sums of the inverses of binomial coefficients’, Integers 5 (2005), Article A22.Google Scholar