gives True if expr is a square-free polynomial or number, and False otherwise.
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 Automatic whether to allow Gaussian integers Modulus 0 modulus for polynomial coefficients
Examplesopen allclose all
SquareFreeQ works over integers:
Specify the variable in a polynomial:
Polynomials over a finite field:
Basic Applications (3)
Number Theory (5)
The central binomial coefficients Binomial[2n,n] are not square free for :
Find the fraction of the first numbers that are square free:
The polynomial p[x]/PolynomialGCD[p[x],p'[x]] is always square free:
The distribution of square-free numbers over integers:
The distribution of square-free numbers over the Gaussian integers:
Properties & Relations (8)
A number that is divisible by a square is not square free:
In the prime factorization of a square-free number, the exponents of primes are all 1:
PrimeNu is equal to PrimeOmega for square-free numbers:
MoebiusMu is zero for non-square-free integers:
Numbers that are prime powers and square free are prime numbers:
The discriminant of a quadratic non-square-free polynomial is 0:
Square factors can be found using FactorSquareFreeList:
Wolfram Research (2007), SquareFreeQ, Wolfram Language function, 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.
Wolfram Language. (2007). SquareFreeQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SquareFreeQ.html