Number theory¶
This chapter describes functions that are of interest in number theory. These functions typically operate on integers. Some of these functions work quite slowly.
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IsPrime
(n)¶ test for a prime number
Param n: integer to test
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IsComposite
(n)¶ test for a composite number
Param n: positive integer
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IsCoprime
(m, n)¶ test if integers are coprime
Param m: positive integer Param n: positive integer Param list: list of positive integers
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IsSquareFree
(n)¶ test for a square-free number
Param n: positive integer
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IsPrimePower
(n)¶ test for a power of a prime number
Param n: integer to test
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NextPrime
(i)¶ generate a prime following a number
Param i: integer value
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IsTwinPrime
(n)¶ test for a twin prime
Param n: positive integer
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IsIrregularPrime
(n)¶ test for an irregular prime
Param n: positive integer
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IsCarmichaelNumber
(n)¶ test for a Carmichael number
Param n: positive integer
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Factors
(x)¶ factorization
Param x: integer or univariate polynomial
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IsAmicablePair
(m, n)¶ test for a pair of amicable numbers
Param m: positive integer Param n: positive integer
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Factor
(x)¶ factorization, in pretty form
Param x: integer or univariate polynomial
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Divisors
(n)¶ number of divisors
Param n: positive integer
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DivisorsSum
(n)¶ the sum of divisors
Param n: positive integer
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ProperDivisors
(n)¶ the number of proper divisors
Param n: positive integer
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ProperDivisorsSum
(n)¶ the sum of proper divisors
Param n: positive integer
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Moebius
(n)¶ the Moebius function
Param n: positive integer
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CatalanNumber
(n)¶ return the
n
-th Catalan NumberParam n: positive integer
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FermatNumber
(n)¶ return the
n
-th Fermat NumberParam n: positive integer
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HarmonicNumber
(n)¶ return the
n
-th Harmonic NumberParam n: positive integer Param r: positive integer
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StirlingNumber1
(n, m)¶ return the
n,m
-th Stirling Number of the first kindParam n: positive integers Param m: positive integers
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StirlingNumber1
(n, m) return the
n,m
-th Stirling Number of the second kindParam n: positive integer Param m: positive integer
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DivisorsList
(n)¶ the list of divisors
Param n: positive integer
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SquareFreeDivisorsList
(n)¶ the list of square-free divisors
Param n: positive integer
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MoebiusDivisorsList
(n)¶ the list of divisors and Moebius values
Param n: positive integer
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SumForDivisors
(var, n, expr)¶ loop over divisors
Param var: atom, variable name Param n: positive integer Param expr: expression depending on var
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RamanujanSum
(k, n)¶ compute the Ramanujan’s sum
Param k: positive integer Param n: positive integer This function computes the Ramanujan’s sum, i.e. the sum of the
n
-th powers of thek
-th primitive roots of the unit:\[\sum_{l=1}^k\frac{\exp(2ln\pi\imath)}{k}\]where \(l\) runs thought the integers between 1 and
k-1
that are coprime to \(l\). The computation is done by using the formula in T. M. Apostol, <i>Introduction to Analytic Theory</i> (Springer-Verlag), Theorem 8.6.Todo
check the definition
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PAdicExpand
(n, p)¶ p-adic expansion
Param n: number or polynomial to expand Param p: base to expand in
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IsQuadraticResidue
(m, n)¶ functions related to finite groups
Param m: integer Param n: odd positive integer
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GaussianFactors
(z)¶ factorization in Gaussian integers
Param z: Gaussian integer
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GaussianNorm
(z)¶ norm of a Gaussian integer
Param z: Gaussian integer
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IsGaussianUnit
(z)¶ test for a Gaussian unit
Param z: a Gaussian integer
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IsGaussianPrime
(z)¶ test for a Gaussian prime
Param z: a complex or real number
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GaussianGcd
(z, w)¶ greatest common divisor in Gaussian integers
Param z: Gaussian integer Param w: Gaussian integer