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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

A $ p$-adic supercongruence conjecture of van Hamme

Author(s): Eric Mortenson
Journal: Proc. Amer. Math. Soc. 136 (2008), 4321-4328.
MSC (2000): Primary 33C20; Secondary 11S80
Posted: June 11, 2008
MathSciNet review: 2431046
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Abstract | References | Similar articles | Additional information

Abstract: In this paper we prove a $ p$-adic supercongruence conjecture of van Hamme by placing it in the context of the Beukers-like supercongruences of Rodriguez-Villegas. This conjecture is a $ p$-adic analog of a formula of Ramanujan.

References:

[A]
S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, Symbolic computation, number theory, special functions, physics and combinatorics. Dev. Math., 4, Kluwer, Dordrecht (2001). MR 1880076 (2003i:33025)

[AO]
S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number supercongruences, J. Reine Angew. Math., 518 (2000), pages 187-212. MR 1739404 (2001c:11057)

[B]
W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 32, Cambridge University Press (1935). MR 0185155 (32:2625)

[Be1]
F. Beukers, Some congruences for Apéry numbers, J. Number Theory, 21 (1985), pages 141-155. MR 808283 (87g:11032)

[Be2]
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory, 25 (1987), pages 201-210. MR 873877 (88b:11002)

[COV]
P. Candelas, X. de la Ossa, and F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields I, http://xxx.lanl.gov/abs/hep-th/0012233.

[G]
J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301 (1987), pages 77-101. MR 879564 (88e:11122)

[I]
T. Ishikawa, On Beukers' congruence, Kobe J. Math., 6 (1989), pages 49-52. MR 1023525 (90i:11001)

[K]
T. Kilbourn, An extension of the Apéry number supercongruences, Acta Arith., 123 (2006), no. 4, pages 335-348. MR 2262248 (2007e:11049)

[McO]
D. McCarthy and R. Osburn, A p-adic analogue of a formula of Ramanujan, preprint.

[M1]
E. Mortenson, A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function, Journal of Number Theory, 99 (2003), pages 139-147. MR 1957248 (2004e:11089)

[M2]
E. Mortenson, Supercongruences between Truncated $ {}_2F_1$ Hypergeometric Functions and Their Gaussian Analogs, Trans. Amer. Math. Soc., 335 (2003), pages 987-1007. MR 1938742 (2003i:11119)

[M3]
E. Mortenson, Supercongruences for the truncated $ {}_{n+1}F_{n}$ hypergeometric series with applications to certain weight three newforms, Proc. Amer. Math. Soc., 133 (2005), pages 321-330. MR 2093051 (2005f:11080)

[Mo]
F. Morley, Note on the congruence $ 2^{4n}\equiv(-1)^n (2n)!/(n!)^2$, where $ 2n+1$ is prime, Annals of Math., 9 (1895), pages 168-170.

[O1]
K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc., 350 (1998), pages 1205-1223. MR 1407498 (98e:11141)

[O2]
K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-Series, CBMS Regional Conference Series in Mathematics, 102, Amer. Math. Soc., Providence, RI, 2004. MR 2020489 (2005c:11053)

[FRV]
F. Rodriguez-Villegas, Hypergeometric Families of Calabi-Yau Manifolds. Calabi-Yau varieties and mirror symmerty, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003. MR 2019156 (2005b:11086)

[WW]
E.T. Whittaker and G.N. Watson, Modern Analysis, fourth ed., Cambridge University Press, 1927.

[vH]
L. van Hamme, Some conjectures concerning partial sums of generalized hypergeometric series. $ p$-adic functional analysis (Nijmegen, 1996), pages 223-236, Lecture Notes in Pure and Appl. Math., 192, Dekker, 1997. MR 1459212 (98k:33011)

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Additional Information:

Eric Mortenson
Affiliation: Department of Mathematics, Penn State University, University Park, Pennsylvania 16802
Email: mort@math.psu.edu

DOI: 10.1090/S0002-9939-08-09389-1
PII: S 0002-9939(08)09389-1
Received by editor(s): September 18, 2007,
Received by editor(s) in revised form: October 22, 2007
Posted: June 11, 2008
Communicated by: Ken Ono
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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