Abstract
In this report we will discuss special congruences. We explain how the congruences arise from formal groups and then we give some examples.
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7.References
K. Alladi and M.L. Robinson: On certain values of the logarithm, Lecture Notes 751, 1–9.
R. Apéry: Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11–13.
A.O.L. Atkin and H.P.F. Swinnerton-Dyer: Modular forms on noncongruence subgroups, Proc. of Symposia in Pure Math., A.M.S. 19 (1971), 1–25.
F.Beukers: Arithmetical properties of Picard-Fuchs equations, Séminaire de théorie des nombres, Paris 82–83, Birkhäuser Boston, 1984, 33–38.
F. Beukers: Some congruences for the Apéry numbers, J. Number Theory 21 (1985), 141–150.
F. Beukers: Another congruence for the Apéry numbers, J. Number Theory 25 (1987), 201–210.
F. Beukers and J. Stienstra: On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Annalen 271 (1985), 293–304.
L. Carlitz: Advanced problem 4268, A.M.M. 62 (1965) p. 186 and A.M.M. 63 (1956) 348–350.
S. Chowla, J. Cowles and M. Cowles: Congruence properties of Apéry numbers, J. Number theory 12 (1980), 188–190.
S. Chowla, B. Dwork and R.J. Evans: On the mod p 2 determination of , J. Number theory 24 (1986), 188–196.
M.J. Coster: Generalisation of a congruence of Gauss, J. Number theory 29 (1988), 300–310.
M.J. Coster: Supercongruences, [Thesis] Univ. of Leiden, the Netherlands, 1988.
M.J. Coster and L. van Hamme: Supercongruences of Atkin and Swinnerton-Dyer type for Legendre polynomials, to appear in J. of Number Theory in 1990.
J. Diamond: The p-adic log gamma function and p-adic Euler constants, Trans. Amer. Math. Soc. 233 (1977), 321–337.
C.F. Gauss: Arithmetische Untersuchungen (Disquisitiones arithmeticae), [Book] Chelsea Publishing Company Bronx, New York, reprinted 1965.
I. Gessel: Some congruences for Apéry numbers, J. Number theory 14 (1982), 362–368.
B. Gross and M. Koblitz: Gauss sums and the p-adic Γ-function, Ann. Math. 109 (1979), 569–581.
L. van Hamme: The p-adic gamma function and congruences of Atkin and Swinnerton-Dyer, Groupe d'étude d'analyse ultramétrique, 9e année 81/82, Fasc. 3 no. J17-6p.
L. van Hamme: Proof of a conjecture of Beukers on Apéry numbers, Proceedings of the conference of p-adic analysis, Hengelhoef, Belgium (1986), 189–195.
M. Hazewinkel: Formal groups and applications, [Book] Academic Press, New York, 1978.
Y. Mimura: Congruence properties of Apéry numbers, J. Number theory 16 (1983), 138–146.
W.H. Schikhof: Ultrametric calculus, [Book] Cambridge University Press, Cambridge, 1984.
H. Weber: Lehrbuch der Algebra, [Book] dritter dand, Friedrich Vieweg und Sohn, Braunschweig, 1908.
P.T. Young: Further congruences for the Apéry numbers, to appear, 1989.
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Coster, M.J. (1990). Supercongruences. In: Baldassarri, F., Bosch, S., Dwork, B. (eds) p-adic Analysis. Lecture Notes in Mathematics, vol 1454. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091139
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DOI: https://doi.org/10.1007/BFb0091139
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