منابع مشابه
Computable p–adic Numbers
In the present work the notion of the computable (primitive recursive, polynomially time computable) p–adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p–adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterizatio...
متن کاملNotes on p-adic numbers
as one can check using induction on l. The usual absolute value function |x| satisfies these conditions with the ordinary triangle inequality (4). If N(x) = 0 when x = 0 and N(x) = 1 when x 6= 0, then N(x) satisfies these conditions with the ultrametric version of the triangle inequality. For each prime number p, the p-adic absolute value of a rational number x is denoted |x|p and defined by |x...
متن کاملComputations with p-adic numbers
This document contains the notes of a lecture I gave at the “Journées Nationales du Calcul Formel” (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of p-adic numbers and compare them and second, we introduce a general framework for studying precision issues and ap...
متن کاملp - adic numbers , LTCC 2010
The following is a proof which is independent of this characterisation. First assume that ‖ ‖ is non-archimedean. Let x, y ∈ K. Using that ‖ ‖ extends | | we then obtain |x + y| = ‖x + y‖ ≤ max{‖x‖, ‖y‖} = max{|x|, |y|} which shows that | | is non-archimedean. Now assume that | | is non-archimedean. Let x, y ∈ K̂. Let ε > 0. Since K is dense in K̂ there exist u, v ∈ K such that ‖x − u‖ < ε and ‖y...
متن کاملComplementary Bell Numbers and p-adic Series
In this article, we generalize a result of Murty on the non-vanishing of complementary Bell numbers and irrationality of a p-adic series. This generalization leads to a sequence of polynomials. We partially answer the question of existence of an integral zero of those polynomials.
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1981
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-39-2-181-191