OpenMath Content Dictionary: integer2

Canonical URL:

http://www.openmath.org/cd/modint.ocd

CD File:

integer2.ocd

CD as XML Encoded OpenMath:

integer2.omcd

Defines:

class, divides, eqmod, euler, modulo_relation, neqmod, ord

Date:

2004-07-11

Version:

0

Review Date:

2006-07-11

Status:

experimental

Uses CD:

alg1, arith1, logic1, quant1, set1, setname1, setname2, relation1, fns1, interval1, integer1

This CD holds a collection of basic modular arithmetic for integers.

modulo_relation

Description:

This symbol represents a univariate function, whose argument should be an integer. When applied to an integer m, it denotes the equivalence relation of being equal modulo m on Z.

Signatures:

sts

divides

Description:

This symbol represents a bivariate Boolean function, whose arguments should be integers. When applied to integers a and b, it denotes the property that a divides b.

Commented Mathematical property (CMP):

For two integers a and b, the number a divides b if and only, in the magma Z with multiplication, a is a left divisor of b.

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA><OMS cd="logic1" name="equivalent"/>
     <OMA><OMS cd="integer2" name="divides"/>
          <OMV name="a"/><OMV name="b"/>
     </OMA>
     <OMA><OMS cd="magma1" name="left_divides"/>
          <OMA><OMS cd="magma1" name="magma"/>
               <OMS cd="setname1" name="Z"/>
               <OMS cd="arith1" name="times"/>
          </OMA>
          <OMV name="a"/><OMV name="b"/>
     </OMA>
</OMA>
</OMOBJ>

equivalent (divides ( a, b) , left_divides (magma (Z, times) , a, b) )

$$a|b\equiv \mathrm{left\_divides}(\mathrm{magma}(\mathbb{Z},\times ),a,b)$$

Signatures:

sts

eqmod

Description:

This symbol represents a Boolean valued trivariate function, whose arguments should be integers. When applied to integers a, b, m, it denotes the Boolean evalue of the assertion that a and b are equal modulo m.

Signatures:

sts

neqmod

Description:

This symbol represents a Boolean valued trivariate function, whose arguments should be integers. When applied to integers a, b, m, it denotes the Boolean evalue of the assertion that a and b are not equal modulo m.

Signatures:

sts

class

Description:

This symbol represents a bivariate function, whose arguments should be integers. If a, m are integers, then class(a,m) denotes the residue class a mod m in setname2.Zm.

Signatures:

sts

euler

Description:

This symbol denotes the univariate Euler totient function. If m is an integer, then euler(m) denotes the order of the multiplicative group of invertible elements in the residue class ring Z/mZ.

Signatures:

sts

ord

Description:

This symbol denotes a binary function. Its first argument shoud be a prime number p, the second an integer n. When applied to p and n, it represents the highest power of p occurring in a factorization of n.

Example:

There are two factors 2 in 60:

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> 
<OMA><OMS name="ord" cd="integer2"/>
    <OMA><OMS name="ord" cd="integer2"/>
         <OMI>2</OMI>  <OMI>60</OMI>
     </OMA>
     <OMI>2</OMI>
</OMA>
</OMOBJ>

ord (ord (2, 60) , 2)

$$\mathrm{ord}(\mathrm{ord}(2,60),2)$$

Signatures:

sts