Square-free number

A natural number is called square-free if there is no square number other than one that divides this number . In other words, no prime number occurs more than once in the unambiguous prime factorization of a square-free number . ${\displaystyle n=p_{1}\cdots p_{k}}$

For example, the number 6 = 2 * 3 is square-free while 54 = 2 * 3 ² * 3 is not square-free. The first 20 square free numbers are

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, ... (sequence A005117 in OEIS )

properties

The Möbius function at this point is not equal to 0 if and only if it is square-free. ${\displaystyle \mu (n)}$ ${\displaystyle n}$ ${\displaystyle n}$

From the main theorem about finitely generated Abelian groups it immediately follows that a finite Abelian group with square-free order is always cyclic .

A number is square-free if and only if the remainder class ring is reduced , that is, if it contains no nilpotent element apart from zero . ${\displaystyle n}$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$

The asymptotic probability that a randomly chosen number is square-free is , where is the Riemann ζ-function . This means: The probability that a uniformly distributed selected natural number is square-free converges for against . ${\displaystyle {\tfrac {1}{\zeta (2)}}={\tfrac {6}{\pi ^{2}}}\approx 61\,\%}$ ${\displaystyle \zeta }$ ${\displaystyle \{1,\dots ,N\}}$ ${\displaystyle N\rightarrow \infty }$ ${\displaystyle {\tfrac {1}{\zeta (2)}}}$

general definition

An element of a factorial ring other than 0 is called square-free if all non-zero exponents are equal to 1 in its prime factorization, which is unique apart from the sequence and multiplication with units of the ring (where one unit of the ring is) . ${\displaystyle x}$ ${\displaystyle x=\varepsilon \cdot p_{1}^{\alpha _{1}}\cdot \ldots \cdot p_{k}^{\alpha _{k}}}$ ${\displaystyle \varepsilon }$ ${\displaystyle \alpha _{i}}$

Let be and the formal derivation, then is square-free if is. Thus for anything the polynomial is always square-free. ${\displaystyle P(x)\in K[X]}$ ${\displaystyle P'(x)}$ ${\displaystyle P(x)}$ ${\displaystyle {\text{ggT}}(P(x),P'(x))=1}$ ${\displaystyle P(x)}$ ${\displaystyle P(x)/{\text{ggT}}(P(x),P'(x))}$

literature

• Paulo Ribenboim: The world of prime numbers . 2nd Edition. Springer Verlag, 2011, ISBN 978-3-642-18078-1 .

Web links

• Eric W. Weisstein : Square Free Number . In: MathWorld (English).