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p-adic Arithmetic

Stany De Smedt
Organization: KBC Bank & Verzekeringen
Journal / Anthology

The Mathematica Journal
Year: 2004
Volume: 9
Issue: 2
Page range: 349-357

Introduction; p-adic Order and Valuation; Conversion to p-adics; Basic p-adic Arithmetic; Booleans and Selectors on p-adic Rationals; Function Expansion; VolkenbornIntegral; p-adic Newton Method; References; Additional Material

The p-adic numbers were introduced by K. Hensel in 1908 in his book Theorie der algebra´schen Zahlen, Leipzig, 1908. In this article we present a package that does p-adic calculations using Mathematica. It allows addition, subtraction, multiplication, and division of p-adic numbers written in Hensel expansion. Functions, such as sqrt, log, exp, sin, cos, sinh, and cosh, are defined using their power series expansion. We also calculate the coefficients of a continuous function with respect to the Mahler and Vanderput bases, and implement the p-adic Newton method.

*Mathematics > Number Theory