IsSquareFree

Tests whether n has no repeated prime factors.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaSquare-free integer↗MathWorldSquarefree↗WikidataQ50706↗OEISA005117↗Wolfram LanguageSquareFreeQ!↗

IsSquareFree(n)tests whether $n$ has no repeated prime factor.

SageMathis_squarefree(n)Juliais_squarefree(ZZ(n))Rustis_square_free(n)

IsSquareFree(f)for a polynomial $f$, whether it has no repeated irreducible factor.

SageMathis_squarefree(f)Juliais_squarefree(ZZ(f))Rustis_square_free(f)

IsSquareFree(f, x)whether $f$ has no repeated factor as a polynomial in $x$ alone.

Typed signatures

Type	Package	Applies
(expression | number | quadratic_integer, any*) -> boolean	number-theoryreplaces compute-engine	always
(integer) -> boolean	compute-engine	always

Domain: Number theory

Details

• An integer is squarefree if it is divisible by no perfect square other than 1.
• Every prime is squarefree; 1 is vacuously squarefree.
• Equivalent to $\mu(n)\neq0$. See MoebiusMu.
• Returns False unless $n$ is provably squarefree.

Examples

5 more kept as data, checked against the oracles

Basic #

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In$\mathrm{IsSquareFree}(10)$Out$\mathrm{True}$

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In$\mathrm{IsSquareFree}(4)$Out$\mathrm{False}$

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In$\mathrm{IsSquareFree}(12)$Out$\mathrm{False}$

Every prime is squarefree

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In$\mathrm{IsSquareFree}(17)$Out$\mathrm{True}$

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In$\mathrm{IsSquareFree}(20)$Out$\mathrm{False}$

Scope #

Threads element-wise over a list

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In$\mathrm{IsSquareFree}(\bigl\lbrack10, 4, 12\bigr\rbrack)$Out$\bigl\lbrack\mathrm{True}, \mathrm{False}, \mathrm{False}\bigr\rbrack$

Gaussian integers: $3+2i$ is a Gaussian prime

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In$\mathrm{IsSquareFree}(3+2i)$Out$\mathrm{True}$

GaussianIntegers -> True: $2=-i(1+i)^2$

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In$\mathrm{IsSquareFree}(2, \mathrm{Rule}(\mathrm{Over}, \mathrm{GaussianIntegers}))$Out$\mathrm{False}$

Rationals: numerator and denominator are both squarefree

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In$\mathrm{IsSquareFree}(\frac{2}{3})$Out$\mathrm{True}$

Polynomials: $x^2+6x+6$ has no repeated factor -- decided by $\gcd(f, f')$ via D and compute-engine's own PolynomialGCD

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In$\mathrm{IsSquareFree}(x^2+6x+6)$Out$\mathrm{True}$

Multivariate: $x^3-x^2y=x^2(x-y)$, a repeated factor of $x$

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In$\mathrm{IsSquareFree}(x^3-x^2y)$Out$\mathrm{False}$

Squarefree as a polynomial in $x$ alone -- $y$ is just a coefficient here

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In$\mathrm{IsSquareFree}(xy^2, x)$Out$\mathrm{True}$

…but not in $y$ alone: $y$ appears squared

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In$\mathrm{IsSquareFree}(xy^2, y)$Out$\mathrm{False}$

A 71-digit integer tested exactly

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In$\mathrm{IsSquareFree}(3+10^{70})$Out$\mathrm{True}$

A prime is squarefree

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In$\mathrm{IsSquareFree}(2)$Out$\mathrm{True}$

Also accepts polynomials

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In$\mathrm{IsSquareFree}(x^2-3)$Out$\mathrm{True}$

Applications #

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In$N(\frac{1}{100\,000}\mathrm{Length}(\mathrm{Filter}(1..100\,000, \mathrm{IsSquareFree})))$Out$0.000\,01\mathrm{Length}(\mathrm{Filter}(1..100\,000, \mathrm{IsSquareFree}))$

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In$\mathrm{IsSquareFree}(q)$Out$\mathrm{True}$

Properties #

$n$ is squarefree exactly when $\mu(n)\neq0$. See MoebiusMu

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In$\mathrm{IsSquareFree}(10)=¬(\mathrm{MoebiusMu}(10)=0)$Out$\mathrm{True}$

$210=2\cdot3\cdot5\cdot7$, no repeated prime

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In$\mathrm{IsSquareFree}(210)$Out$\mathrm{True}$

$14=2\cdot 7$, no repeated prime

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In$\mathrm{IsSquareFree}(14)$Out$\mathrm{True}$

Primes are squarefree

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In$\mathrm{IsPrimePower}(17)∧\mathrm{IsSquareFree}(17)$Out$\mathrm{True}$

$(x-1)^2$ has a repeated factor

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In$\mathrm{IsSquareFree}(x^2-2x+1, x)$Out$\mathrm{False}$

A square of an integer is not squarefree, given $a\in\mathbb Z$

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In$\mathrm{Simplify}(\mathrm{IsSquareFree}(a^2), a∈ℤ)$Out$\mathrm{False}$

Expanding does not hide the repeated factors

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In$\mathrm{IsSquareFree}(\mathrm{Expand}((x+1)^3(x+2)^2(x+3)^2))$Out$\mathrm{False}$

Possible issues #

1 is vacuously squarefree: it has no repeated prime factors

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In$\mathrm{IsSquareFree}(1)$Out$\mathrm{True}$

About

Examples	31
Checked against	Wolfram Language 27 agree2 differSageMath 14 agree1 differ7 errorSymPy 18 agree4 differ3 errormathlib4 10 agree4 errorJulia 12 agree3 errorRust 10 agree4 error
Crosswalk	WikipediaMathWorldWikidataOEISWolfram LanguageSageMathJuliaRust

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

Every exponent of the factorization is 1.

mappedsympypackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Squarefree on the natural numbers, so Squarefree 0 is false and a negative n fails to typecheck.

See also: MoebiusMu, FactorInteger