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 .
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.
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 .
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 .
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) .
Let be and the formal derivation, then is square-free if is. Thus for anything the polynomial is always square-free.
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).