Results in Mathematics

, Volume 15, Issue 3–4, pp 324–334 | Cite as

P-Adic and Non-Archimedean Product Representations

  • Arnold Knopfmacher
  • John Knopfmacher


Two algorithms are introduced and shown to lead to a unique product representation for a given p-adic integer A with leading coefficient 1, as a product of terms \(1+{1\over a_n}\) where the a n are certain rational numbers. The degree of approximation by the N-th partial product of such terms is investigated, and in some explicit cases the products corresponding to particular types of p-adic integers are characterized. In addition, we consider similar representations for elements of arbitrary complete non-archimedean fields with discrete valuations.

AMS (1980/85) Classification Numbers

11J61 11K41 


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  1. 1.
    E.B. Escott, Rapid method for extracting square roots, Amer. Math. Monthly, 44 (1937), 644–646.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    V. Laohakosol. A characterization of rational numbers by p-adic Ruban continued fractions. J. Austral. Math. Soc, A 39 (1985), 300–305.MathSciNetCrossRefGoogle Scholar
  3. 3.
    K. Mahler. Zur Approximation p-adischer Irrationalzahlen. Nieuw Archief v. Wisk., 18 (1934), 22–34.zbMATHGoogle Scholar
  4. 4.
    A. Oppenheim. The representation of real numbers by infinite series of rationals, Acta Arith., 21 (1972), 391–398.MathSciNetzbMATHGoogle Scholar
  5. 5.
    A.M. Ostrowski. Über einige Verallgemeinerungen des Eulerschen Produktesh., Verh. Naturforsch. Gesellsch. Basel, 40 (1929), 153–204.Google Scholar
  6. 6.
    O. Perron. Irrationalzahlen. Chelsea Publ. Co., 1951.Google Scholar
  7. 7.
    A.A. Ruban. Some metric properties of p-adic numbers. Siberian Math. J., 11 (1970), 178–180.MathSciNetCrossRefGoogle Scholar
  8. 8.
    W.H. Schikhof, Ultrametric Calculus. Cambridge University Press, 1984.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1989

Authors and Affiliations

  • Arnold Knopfmacher
    • 1
  • John Knopfmacher
    • 2
  1. 1.Department of Computational & Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.Department of MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa

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