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SquareFreeQ
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SquareFreeQ

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SquareFreeQ[expr]

gives True if expr is a square-free polynomial or number, and False otherwise.

✖
SquareFreeQ[expr,vars]

gives True if expr is square free with respect to the variables vars.

Details and Options

  • SquareFreeQ is typically used to test whether a number or a polynomial is square free.
  • An integer n is square free if it is divisible by no perfect square other than 1.
  • SquareFreeQ[expr] returns False unless expr is manifestly square free.
  • With the setting GaussianIntegers->True, SquareFreeQ tests whether expr is Gaussian square free.
  • For integers m and n, SquareFreeQ[m+I n] automatically works over Gaussian integers.
  • The following options can be given:
  • GaussianIntegers Automaticwhether to allow Gaussian integers
    Modulus 0modulus for polynomial coefficients

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Test whether a number is square free:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-f74ygq
Out[1]=1

The number 4 is not square free:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-irhb4t
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

SquareFreeQ works over integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-rme16t
Out[1]=1

Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-lfc27m
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-1kr17a
Out[2]=2

Rational numbers:

In[4]:=4
✖
https://wolfram.com/xid/0d6d8g8g1f-wu79y6
Out[4]=4

Univariate polynomials:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-kfw85e
Out[1]=1

Multivariate polynomials:

In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-8xg5ug
Out[2]=2

Specify the variable in a polynomial:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-27s9gz
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-dipf6m
Out[2]=2

Polynomials over a finite field:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-gdwau
Out[1]=1

Test for large integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-1cuxzp
Out[1]=1

Options  (2)Common values & functionality for each option

GaussianIntegers  (1)

Test whether 2 is square free over integers:

In[4]:=4
✖
https://wolfram.com/xid/0d6d8g8g1f-10e1vr
Out[4]=4

Gaussian integers:

In[6]:=6
✖
https://wolfram.com/xid/0d6d8g8g1f-s7a4h8
Out[6]=6

Modulus  (1)

Test whether is square free over integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-9p8cp6
Out[1]=1

Integers modulo 3:

In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-tkpubc
Out[2]=2

Applications  (8)Sample problems that can be solved with this function

Basic Applications  (3)

Highlight square-free numbers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-dpsqd1
Out[1]=1

Generate random square-free integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-s7dgp
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-bqtx35
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6d8g8g1f-vjnd2
Out[3]=3

Square-free Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-y8p5c
Out[1]=1

Number Theory  (5)

The central binomial coefficients Binomial[2n,n] are not square free for :

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-dwydb7
Out[1]=1

Find the fraction of the first numbers that are square free:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-5atbur
Out[1]=1

The result is close to :

In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-ge9z9w
Out[2]=2

The polynomial p[x]/PolynomialGCD[p[x],p'[x]] is always square free:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-dx1xo6
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-bntzft
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6d8g8g1f-1b1jqr
Out[3]=3

The distribution of square-free numbers over integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-hi0e4n
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-iul1cx

Plot the distribution:

In[3]:=3
✖
https://wolfram.com/xid/0d6d8g8g1f-d4lpp
Out[3]=3

The distribution of square-free numbers over the Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-9om2p7
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-1ullvw

Plot the distribution:

In[3]:=3
✖
https://wolfram.com/xid/0d6d8g8g1f-jrlfrw
Out[3]=3

Properties & Relations  (8)Properties of the function, and connections to other functions

A number that is divisible by a square is not square free:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-d29ykm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-1rmlvd
Out[2]=2

In the prime factorization of a square-free number, the exponents of primes are all 1:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-c5x9z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-m1yhtl
Out[2]=2

PrimeNu is equal to PrimeOmega for square-free numbers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-dxtjw6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-cwp7f3
Out[2]=2

MoebiusMu is zero for non-square-free integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-ibqbqf
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-ncxg8e
Out[2]=2

Numbers that are prime powers and square free are prime numbers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-8pn4ph
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-jr9o01
Out[2]=2

The discriminant of a quadratic non-square-free polynomial is 0:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-b2ce5w
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-bm1bsn
Out[2]=2

Square factors can be found using FactorSquareFreeList:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-ef7qfk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-ipkc0i
Out[2]=2

Simplify symbolic expressions:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-ob9vuq
Out[1]=1

Neat Examples  (3)Surprising or curious use cases

Plot the prime numbers that are the sum of three squares:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-h7zwj2
Out[1]=1

Square-free Gaussian integers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-epjx15
Out[1]=1

Plot the Ulam spiral of square-free numbers:

In[1]:=1
✖
https://wolfram.com/xid/0d6d8g8g1f-qu5xoh
In[2]:=2
✖
https://wolfram.com/xid/0d6d8g8g1f-fph21d
Out[2]=2
Wolfram Research (2007), SquareFreeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SquareFreeQ.html.
✖
Wolfram Research (2007), SquareFreeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SquareFreeQ.html.

Text

Wolfram Research (2007), SquareFreeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SquareFreeQ.html.

✖
Wolfram Research (2007), SquareFreeQ, Wolfram Language function, https://reference.wolfram.com/language/ref/SquareFreeQ.html.

CMS

Wolfram Language. 2007. "SquareFreeQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SquareFreeQ.html.

✖
Wolfram Language. 2007. "SquareFreeQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SquareFreeQ.html.

APA

Wolfram Language. (2007). SquareFreeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SquareFreeQ.html

✖
Wolfram Language. (2007). SquareFreeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SquareFreeQ.html

BibTeX

@misc{reference.wolfram_2025_squarefreeq, author="Wolfram Research", title="{SquareFreeQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SquareFreeQ.html}", note=[Accessed: 10-July-2025 ]}

✖
@misc{reference.wolfram_2025_squarefreeq, author="Wolfram Research", title="{SquareFreeQ}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SquareFreeQ.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_squarefreeq, organization={Wolfram Research}, title={SquareFreeQ}, year={2007}, url={https://reference.wolfram.com/language/ref/SquareFreeQ.html}, note=[Accessed: 10-July-2025 ]}

✖
@online{reference.wolfram_2025_squarefreeq, organization={Wolfram Research}, title={SquareFreeQ}, year={2007}, url={https://reference.wolfram.com/language/ref/SquareFreeQ.html}, note=[Accessed: 10-July-2025 ]}