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Abstract

The p-adic arithmetic allows error-free representation of fractions and error-free arithmetic
using fractions . In this tutorial, we describe infinite- precision p -adic arithmetic which is more
suitable for software implementations and finite-precision p-adic arithmetic which is more suitable
for hardware implementations. The finite-precision p-adic representation is also called
Hensel code which has certain interesting properties and some open problems for research.

1 Introduction

A p- adic number α can be uniquely written in the form

where each of ∈ [0, p − 1] and the p-adic norm of the number α is defined as . Note
that the series

converges to in the p-adic norm. Now, as an example, consider the power series expansion

Since converges to , we have

Taking p = 5, we obtain 5-adic expansion of  , which can be written in the form

There is a one-to-one correspondence between the power series expansion

and the short representation , where only the coefficients of the powers of p are
shown. We can use the p-adic point as a device for displaying the sign of n .

For example,

2 Representation of Negative Numbers

If

then

where and for i > n. Thus, for example,

However, watch for leading zeros , they remain unchanged:

3 Representation of Integers

Since a positive integer h can be expressed in exactly one way as the sum of powers of a prime p,
i.e.,

with ∈ [0, p − 1], there is essentially no difference between p-adic and p-ary representation of h.
The only difference is that the digits in the p -adic representation are written in reverse order. For
example,

4 Representation of Rational Numbers

If α is a rational number , then it has a repeating pattern of   in its p-adic expansion, i.e., it is of
the form

For example, , and , etc. Let α have the p-adic expansion

where and p divides neither nor . The p-adic expansion for is

and thus

In other words, we compute by

Next, we use

where and p divides neither nor . The p-adic expansion for is

and so

We continue this process until the period is exhibited. Let α = 2/15 and p = 5. Thus,

and n = -1. The 5-adic expansion of 2/15 is found as

which gives us .

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