# OpenMath Content Dictionary: integer2

Canonical URL:
http://www.openmath.org/cd/integer2.ocd
CD Base:
http://www.openmath.org/cd
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 (Revision 1)
Review Date:
2006-07-11
Status:
experimental

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

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## 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):
$a|b\equiv \mathrm{left_divides}\left(\mathrm{magma}\left(\mathbb{Z},×\right),a,b\right)$
Signatures:
sts

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## 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

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## 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

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## 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

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## 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

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## 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:
$\mathrm{ord}\left(\mathrm{ord}\left(2,60\right),2\right)$
Signatures:
sts

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