In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 3². The smallest positive square-free numbers are

Square-free factorization

Every positive integer ${\displaystyle n}$ can be factored in a unique way as

where the ${\displaystyle q_{i}}$ different from one are square-free integers that are pairwise coprime. This is called the square-free factorization of n.

To construct the square-free factorization, let

be the prime factorization of ${\displaystyle n}$, where the ${\displaystyle p_{j}}$ are distinct prime numbers. Then the factors of the square-free factorization are defined as

An integer is square-free if and only if ${\displaystyle q_{i}=1}$ for all ${\displaystyle i>1}$. An integer greater than one is the ${\displaystyle k}$th power of another integer if and only if ${\displaystyle k}$ is a divisor of all ${\displaystyle i}$ such that ${\displaystyle q_{i}\neq 1.}$

The use of the square-free factorization of integers is limited by the fact that its computation is as difficult as the computation of the prime factorization. More precisely every known algorithm for computing a square-free factorization computes also the prime factorization. This is a notable difference with the case of polynomials for which the same definitions can be given, but, in this case, the square-free factorization is not only easier to compute than the complete factorization, but it is the first step of all standard factorization algorithms.

Square-free factors of integers

The radical of an integer is its largest square-free factor, that is ${\displaystyle \textstyle \prod _{i=1}^{k}q_{i}}$ with notation of the preceding section. An integer is square-free if and only if it is equal to its radical.

Every positive integer ${\displaystyle n}$ can be represented in a unique way as the product of a powerful number (that is an integer such that is divisible by the square of every prime factor) and a square-free integer, which are coprime. In this factorization, the square-free factor is ${\displaystyle q_{1},}$ and the powerful number is ${\displaystyle \textstyle \prod _{i=2}^{k}q_{i}^{i}.}$

The square-free part of ${\displaystyle n}$ is ${\displaystyle q_{1},}$ which is the largest square-free divisor ${\displaystyle k}$ of ${\displaystyle n}$ that is coprime with ${\displaystyle n/k}$. The square-free part of an integer may be smaller than the largest square-free divisor, which is ${\displaystyle \textstyle \prod _{i=1}^{k}q_{i}.}$

Any arbitrary positive integer ${\displaystyle n}$ can be represented in a unique way as the product of a square and a square-free integer:

In this factorization, ${\displaystyle m}$ is the largest divisor of ${\displaystyle n}$ such that ${\displaystyle m^{2}}$ is a divisor of ${\displaystyle n}$.

In summary, there are three square-free factors that are naturally associated to every integer: the square-free part, the above factor ${\displaystyle k}$, and the largest square-free factor. Each is a factor of the next one. All are easily deduced from the prime factorization or the square-free factorization: if

are the prime factorization and the square-free factorization of ${\displaystyle n}$, where ${\displaystyle p_{1},\ldots ,p_{h}}$ are distinct prime numbers, then the square-free part is The square-free factor such the quotient is a square is and the largest square-free factor is

For example, if ${\displaystyle n=75600=2^4\cdot <!--\; \cdot \; -->3^3\cdot <!--\; \cdot \; -->{}^{}}$Zdroj:https://en.wikipedia.org?pojem=Square-free_integer
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