Overview of the NumberTheory Package
Calling Sequence
Description
List of NumberTheory Package Commands
Related Commands
Examples
Compatibility
NumberTheory:-command(arguments)
command(arguments)
The NumberTheory package contains commands used to investigate the properties of the natural numbers and integers (see number theory).
Each command in the NumberTheory package can be accessed by using either the long form or the short form of the command name in the calling sequence.
The long form, NumberTheory:-command, is always available. The short form can be used after loading the package.
The following is a list of commands in the NumberTheory package.
AreCoprime
test whether a sequence of numbers is relatively prime
CalkinWilfSequence
compute the nth term in the Calkin-Wilf sequence
CarmichaelLambda
Carmichael's lambda function
ChineseRemainder
generalized Chinese remainder algorithm
ContinuedFraction
continued fraction expansion
ContinuedFractionPolynomial
simple continued fraction expansions for real roots of a rational polynomial
CyclotomicPolynomial
minimal polynomials of primitive roots of unity with rational coefficients
Divisors
the set of positive divisors of an integer
FactorNormEuclidean
factorization of integers in quadratic norm-Euclidean fields
HomogeneousDiophantine
solution to Minkowski's linear forms
ImaginaryUnit
modular square root of -1
InhomogeneousDiophantine
inhomogeneous Diophantine approximation
IntegralBasis
integral base of an algebraic number field
InverseTotient
inverse of Euler's totient function
IsCyclotomicPolynomial
test whether a polynomial is cyclotomic
IsMersenne
test whether a number is a Mersenne number
IsSquareFree
test whether an integer is square free
IthFermat
Fermat numbers
IthMersenne
Mersenne exponents
JacobiSymbol
generalized Legendre symbol
JordanTotient
Jordan's totient function
KroneckerSymbol
generalized Jacobi symbol
Landau
compute the Landau g function
LargestNthPower
largest integer power divisor of a number
LegendreSymbol
quadratic residuosity
ModExtendedGCD
solutions to the modulo n extended GCD problem
ModularLog
discrete logarithm under modular arithmetics
ModularRoot
modular root
ModularSquareRoot
modular square root
Moebius
Möbius function
MultiplicativeOrder
order of a number under modular multiplication
NearestLatticePoint
solution to the nearby lattice point problem
NextSafePrime
least safe prime greater than a number
NumberOfIrreduciblePolynomials
number of monic irreducible polynomials
NumberOfPrimeFactors
number of prime factors counted with multiplicity
PrimeCounting
number of prime numbers less than a number
PrimeFactors
prime factors of an integer
PrimitiveRoot
primitive root modulo n
PseudoPrimitiveRoot
pseudo primitive root modulo n
QuadraticResidue
quadratic residuosity of a number
Radical
radical of an integer
RepeatingDecimal
rational number in repeating decimal form
RootsOfUnity
modular roots of unity
SimplestRational
compute the simplest rational number in a real interval
SumOfDivisors
sum of powers of the divisors
SumOfSquares
solutions to the sum of two squares problem
ThueSolve
solutions to a Thue equation or inequality
Totient
Euler's totient function
The following commands are not in the NumberTheory package, but are closely related to number theory.
bernoulli
Bernoulli polynomials
chrem
Chinese Remainder Algorithm
doublefactorial
compute double factorial
euler
Euler polynomials
factorial
compute factorial
GaussInt[GIgcd]
greatest common divisor of Gaussian integers
ifactor
integer factorization
ifactors
igcd
greatest common divisor of integers
ilcm
least common multiple of integers
iroot
compute integer roots
isolve
solve Diophantine equations for integer solutions
isprime
primality test
isqrt
compute integer square roots
issqr
test for a perfect square
ithprime
determine the ith prime number
nextprime
determine the next largest prime
prevprime
determine the next smallest prime
surd
non-principal root function
withNumberTheory:
Show 37 as a repeated decimal:
RepeatingDecimal37
0.428571&conjugate0;
Show 37 as a continued fraction:
ContinuedFraction37
0+12+13+0
Euler's totient (phi) function is an arithmetic function that counts the positive integers less than or equal to a given value, n, that are coprime to n. The PrimeCounting (or pi) command returns the number of primes less than an integer, n.
Comparing pi(n) with phi(n) for the first forty values for n:
plots:-displayDynamicSystems:-DiscretePlotseqi−0.1,i=2..40,seqPrimeCountingn,n=2..40,style=stem,symbol=soliddiamond,color=Crimson,legend=pi,DynamicSystems:-DiscretePlotseqi+0.1,i=2..40,seqTotientn,n=2..40,style=stem,symbol=solidcircle,color=MidnightBlue,legend=Totient,linestyle=dot,transparency=0.1,size=800,400,axis2=gridlines=thickness=0,color=LightGrey
Two integers are relatively prime (coprime) if the greatest common divisor of the values is 1. The following plot shows the coprimes for the integers 1 to 25:
Statistics:-HeatMapMatrix25,i,j↦`if`AreCoprimei,j,1,0,color=White,Pink
The NumberTheory package was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
Applications
Arithmetic Functions
Prime Numbers
Divisibility
Mersenne Primes
See Also
numtheory(deprecated)
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