square-free number

A square-free number is a natural numberMathworldPlanetmath that contains no powers greater than 1 in its prime factorizationMathworldPlanetmath. In other words, if x is our number, and


is the prime factorization of x into r distinct primes, then ai2 is always false for square-free x.

Note: we assume here that x itself must be greater than 1; hence 1 is not considered square-free. However, one must be alert to the particular context in which “square-free” is used as to whether this is considered the case.

The name derives from the fact that if any ai were to be greater than or equal to two, we could be sure that at least one square divides x (namely, pi2.)

1 Asymptotic Analysis

The asymptotic density of square-free numbers is 6π2 which can be proved by application of a square-free variation of the sieve of EratosthenesMathworldPlanetmathPlanetmath (http://planetmath.org/SieveOfEratosthenes2) as follows:

A(n) =kn[k is squarefree ]

It was shown that the Riemann HypothesisMathworldPlanetmath implies error term O(n7/22+ϵ) in the above [1].


  • 1 R. C. Baker and J. Pintz. The distribution of square-free numbers. Acta Arith., 46:73–79, 1985. http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0535.10045Zbl 0535.10045.
Title square-free number
Canonical name SquarefreeNumber
Date of creation 2013-03-22 11:55:33
Last modified on 2013-03-22 11:55:33
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 17
Author akrowne (2)
Entry type Definition
Classification msc 11A51
Synonym square free number
Synonym square free
Synonym square-free
Synonym squarefree
Related topic MoebiusFunction
Related topic SquareRootsOfRationals