IntegerNumberSystem

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An IntegerNumberSystem is a model for the integers.

0: %

from AbelianMonoid

1: %

from MagmaWithUnit

*: (%, %) -> %

from LeftModule %

*: (Integer, %) -> %

from AbelianGroup

*: (NonNegativeInteger, %) -> %

from AbelianMonoid

*: (PositiveInteger, %) -> %

from AbelianSemiGroup

+: (%, %) -> %

from AbelianSemiGroup

-: % -> %

from AbelianGroup

-: (%, %) -> %

from AbelianGroup

<=: (%, %) -> Boolean

from PartialOrder

<: (%, %) -> Boolean

from PartialOrder

=: (%, %) -> Boolean

from BasicType

>=: (%, %) -> Boolean

from PartialOrder

>: (%, %) -> Boolean

from PartialOrder

^: (%, NonNegativeInteger) -> %

from MagmaWithUnit

^: (%, PositiveInteger) -> %

from Magma

~=: (%, %) -> Boolean

from BasicType

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: % -> OutputForm

from CoercibleTo OutputForm

coerce: Integer -> %

from NonAssociativeRing

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.

D: % -> %

from DifferentialRing

D: (%, NonNegativeInteger) -> %

from DifferentialRing

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

hash: % -> SingleInteger

from SetCategory

hashUpdate!: (HashState, %) -> HashState

from SetCategory

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

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?: % -> Boolean

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

rational: % -> Fraction Integer

rational(n) creates 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

AbelianGroup

AbelianMonoid

AbelianSemiGroup

Algebra %

BasicType

BiModule(%, %)

CancellationAbelianMonoid

canonicalUnitNormal

CharacteristicZero

CoercibleFrom Integer

CoercibleTo OutputForm

CombinatorialFunctionCategory

CommutativeRing

CommutativeStar

Comparable

ConvertibleTo DoubleFloat

ConvertibleTo Float

ConvertibleTo InputForm

ConvertibleTo Integer

ConvertibleTo Pattern Integer

DifferentialRing

EntireRing

EuclideanDomain

GcdDomain

IntegralDomain

LeftModule %

LeftOreRing

Magma

MagmaWithUnit

Module %

Monoid

multiplicativeValuation

NonAssociativeRing

NonAssociativeRng

NonAssociativeSemiRing

NonAssociativeSemiRng

noZeroDivisors

OrderedAbelianGroup

OrderedAbelianMonoid

OrderedAbelianSemiGroup

OrderedCancellationAbelianMonoid

OrderedIntegralDomain

OrderedRing

OrderedSet

PartialOrder

PatternMatchable Integer

PrincipalIdealDomain

RealConstant

RetractableTo Integer

RightModule %

Ring

Rng

SemiGroup

SemiRing

SemiRng

SetCategory

StepThrough

TwoSidedRecip

UniqueFactorizationDomain

unitsKnown