IntegerNumberSystem

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An IntegerNumberSystem is a model for the integers.

* : (%, %) -> %

from Magma

* : (Integer, %) -> %

from AbelianGroup

* : (NonNegativeInteger, %) -> %

from AbelianMonoid

* : (PositiveInteger, %) -> %

from AbelianSemiGroup

+ : (%, %) -> %

from AbelianSemiGroup

- : % -> %

from AbelianGroup

- : (%, %) -> %

from AbelianGroup

0 : () -> %

from AbelianMonoid

1 : () -> %

from MagmaWithUnit

< : (%, %) -> Boolean

from PartialOrder

<= : (%, %) -> Boolean

from PartialOrder

= : (%, %) -> Boolean

from BasicType

> : (%, %) -> Boolean

from PartialOrder

>= : (%, %) -> Boolean

from PartialOrder

D : % -> %

from DifferentialRing

D : (%, NonNegativeInteger) -> %

from DifferentialRing

^ : (%, NonNegativeInteger) -> %

from MagmaWithUnit

^ : (%, PositiveInteger) -> %

from Magma

abs : % -> %

from OrderedRing

addmod : (%, %, %) -> %

addmod(a, b, p), 0<=a, b<p>1, means a+b mod p.

annihilate? : (%, %) -> Boolean

from Rng

antiCommutator : (%, %) -> %

from NonAssociativeSemiRng

associates? : (%, %) -> Boolean

from EntireRing

associator : (%, %, %) -> %

from NonAssociativeRng

base : () -> %

base() returns the base for the operations of IntegerNumberSystem.

binomial : (%, %) -> %

from CombinatorialFunctionCategory

bit? : (%, %) -> Boolean

bit?(n, i) returns true if and only if i-th bit of n is a 1.

characteristic : () -> NonNegativeInteger

from NonAssociativeRing

coerce : % -> %

from Algebra(%)

coerce : Integer -> %

from CoercibleFrom(Integer)

coerce : % -> OutputForm

from CoercibleTo(OutputForm)

commutator : (%, %) -> %

from NonAssociativeRng

convert : % -> DoubleFloat

from ConvertibleTo(DoubleFloat)

convert : % -> Float

from ConvertibleTo(Float)

convert : % -> InputForm

from ConvertibleTo(InputForm)

convert : % -> Integer

from ConvertibleTo(Integer)

convert : % -> Pattern(Integer)

from ConvertibleTo(Pattern(Integer))

copy : % -> %

copy(n) gives a copy of n.

dec : % -> %

dec(x) returns x - 1.

differentiate : % -> %

from DifferentialRing

differentiate : (%, NonNegativeInteger) -> %

from DifferentialRing

divide : (%, %) -> Record(quotient : %, remainder : %)

from EuclideanDomain

euclideanSize : % -> NonNegativeInteger

from EuclideanDomain

even? : % -> Boolean

even?(n) returns true if and only if n is even.

expressIdealMember : (List(%), %) -> Union(List(%), "failed")

from PrincipalIdealDomain

exquo : (%, %) -> Union(%, "failed")

from EntireRing

extendedEuclidean : (%, %) -> Record(coef1 : %, coef2 : %, generator : %)

from EuclideanDomain

extendedEuclidean : (%, %, %) -> Union(Record(coef1 : %, coef2 : %), "failed")

from EuclideanDomain

factor : % -> Factored(%)

from UniqueFactorizationDomain

factorial : % -> %

from CombinatorialFunctionCategory

gcd : (%, %) -> %

from GcdDomain

gcd : List(%) -> %

from GcdDomain

gcdPolynomial : (SparseUnivariatePolynomial(%), SparseUnivariatePolynomial(%)) -> SparseUnivariatePolynomial(%)

from GcdDomain

inc : % -> %

inc(x) returns x + 1.

init : () -> %

from StepThrough

invmod : (%, %) -> %

invmod(a, b), 0<=a<b>1, (a, b)=1 means 1/a mod b.

latex : % -> String

from SetCategory

lcm : (%, %) -> %

from GcdDomain

lcm : List(%) -> %

from GcdDomain

lcmCoef : (%, %) -> Record(llcm_res : %, coeff1 : %, coeff2 : %)

from LeftOreRing

leftPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

leftPower : (%, PositiveInteger) -> %

from Magma

leftRecip : % -> Union(%, "failed")

from MagmaWithUnit

length : % -> %

length(a) length of a in digits.

mask : % -> %

mask(n) returns 2^n-1 (an n bit mask).

max : (%, %) -> %

from OrderedSet

min : (%, %) -> %

from OrderedSet

mulmod : (%, %, %) -> %

mulmod(a, b, p), 0<=a, b<p>1, means a*b mod p.

multiEuclidean : (List(%), %) -> Union(List(%), "failed")

from EuclideanDomain

negative? : % -> Boolean

from OrderedRing

nextItem : % -> Union(%, "failed")

from StepThrough

odd? : % -> Boolean

odd?(n) returns true if and only if n is odd.

one? : % -> Boolean

from MagmaWithUnit

opposite? : (%, %) -> Boolean

from AbelianMonoid

patternMatch : (%, Pattern(Integer), PatternMatchResult(Integer, %)) -> PatternMatchResult(Integer, %)

from PatternMatchable(Integer)

permutation : (%, %) -> %

from CombinatorialFunctionCategory

plenaryPower : (%, PositiveInteger) -> %

from NonAssociativeAlgebra(%)

positive? : % -> Boolean

from OrderedRing

positiveRemainder : (%, %) -> %

positiveRemainder(a, b) (where b > 1) yields r where 0 <= r < b and r = a rem b.

powmod : (%, %, %) -> %

powmod(a, b, p), 0<=a, b<p>1, means a^b mod p.

prime? : % -> Boolean

from UniqueFactorizationDomain

principalIdeal : List(%) -> Record(coef : List(%), generator : %)

from PrincipalIdealDomain

quo : (%, %) -> %

from EuclideanDomain

random : % -> %

random(n) creates a random element from 0 to n-1.

rational : % -> Fraction(Integer)

rational(n) creates a rational number (see Fraction Integer).

rational? : % -> Boolean

rational?(n) tests if n is a rational number (see Fraction Integer).

rationalIfCan : % -> Union(Fraction(Integer), "failed")

rationalIfCan(n) creates a rational number, or returns "failed" if this is not possible.

recip : % -> Union(%, "failed")

from MagmaWithUnit

rem : (%, %) -> %

from EuclideanDomain

retract : % -> Integer

from RetractableTo(Integer)

retractIfCan : % -> Union(Integer, "failed")

from RetractableTo(Integer)

rightPower : (%, NonNegativeInteger) -> %

from MagmaWithUnit

rightPower : (%, PositiveInteger) -> %

from Magma

rightRecip : % -> Union(%, "failed")

from MagmaWithUnit

sample : () -> %

from AbelianMonoid

shift : (%, %) -> %

shift(a, i) shift a by i digits.

sign : % -> Integer

from OrderedRing

sizeLess? : (%, %) -> Boolean

from EuclideanDomain

smaller? : (%, %) -> Boolean

from Comparable

squareFree : % -> Factored(%)

from UniqueFactorizationDomain

squareFreePart : % -> %

from UniqueFactorizationDomain

submod : (%, %, %) -> %

submod(a, b, p), 0<=a, b<p>1, means a-b mod p.

subtractIfCan : (%, %) -> Union(%, "failed")

from CancellationAbelianMonoid

symmetricRemainder : (%, %) -> %

symmetricRemainder(a, b) (where b > 1) yields r where -b/2 < r <= b/2.

unit? : % -> Boolean

from EntireRing

unitCanonical : % -> %

from EntireRing

unitNormal : % -> Record(unit : %, canonical : %, associate : %)

from EntireRing

zero? : % -> Boolean

from AbelianMonoid

~= : (%, %) -> Boolean

from BasicType

ConvertibleTo(Float)

PrincipalIdealDomain

ConvertibleTo(Integer)

NonAssociativeSemiRing

IntegralDomain

BiModule(%, %)

ConvertibleTo(InputForm)

canonicalUnitNormal

Rng

CoercibleFrom(Integer)

TwoSidedRecip

SemiRing

EntireRing

unitsKnown

noZeroDivisors

OrderedSet

UniqueFactorizationDomain

SemiGroup

Magma

GcdDomain

LeftModule(%)

NonAssociativeRing

CharacteristicZero

OrderedIntegralDomain

CommutativeRing

Algebra(%)

DifferentialRing

OrderedAbelianMonoid

CombinatorialFunctionCategory

NonAssociativeSemiRng

PartialOrder

CancellationAbelianMonoid

EuclideanDomain

Comparable

RetractableTo(Integer)

OrderedCancellationAbelianMonoid

OrderedRing

CommutativeStar

AbelianMonoid

MagmaWithUnit

RightModule(%)

RealConstant

ConvertibleTo(DoubleFloat)

OrderedAbelianSemiGroup

Module(%)

ConvertibleTo(Pattern(Integer))

CoercibleTo(OutputForm)

SemiRng

Monoid

LeftOreRing

NonAssociativeAlgebra(%)

BasicType

Ring

AbelianSemiGroup

SetCategory

multiplicativeValuation

NonAssociativeRng

PatternMatchable(Integer)

OrderedAbelianGroup

StepThrough

AbelianGroup