Square-free integer


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

10 is square-free, as its divisors greater than 1 are 2, 5, and 10, none of which is square (the first few squares being 1, 4, 9, and 16)
Square-free integers up to 120 remain after eliminating multiples of squares of primes up to √120
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ... (sequence A005117 in the OEIS)

Square-free factorizationEdit

Every positive integer   can be factored in a unique way as

where the   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  , where the   are distinct prime numbers. Then the factors of the square-free factorization are defined as

An integer is square-free if and only if   for all  . An integer greater than one is the  th power of another integer if and only if   is a divisor of all   such that  

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 integersEdit

The radical of an integer is its largest square-free factor, that is   with notation of the preceding section. An integer is square-free if and only if it is equal to its radical.

Every positive integer   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   and the powerful number is  

The square-free part of   is   which is the largest square-free divisor   of   that is coprime with  . The square-free part of an integer may be smaller than the largest square-free divisor, which is  

Any arbitrary positive integer   can be represented in a unique way as the product of a square and a square-free integer:

In this factorization,   is the largest divisor of   such that   is a divisor of  .

In summary, there are three square-free factors that are naturally associated to every integer: the square-free part, the above factor  , 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  , where   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   one has   The square-free part is 7, the square-free factor such that the quotient is a square is 3 ⋅ 7 = 21, and the largest square-free factor is 2 ⋅ 3 ⋅ 5 ⋅ 7 = 210.

No algorithm is known for computing any of these square-free factors which is faster than computing the complete prime factorization. In particular, there is no known polynomial-time algorithm for computing the square-free part of an integer, or even for determining whether an integer is square-free.[1] In contrast, polynomial-time algorithms are known for primality testing.[2] This is a major difference between the arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials (in short, the largest square-free factor of a polynomial is its quotient by the greatest common divisor of the polynomial and its formal derivative).[3]

Equivalent characterizationsEdit

A positive integer   is square-free if and only if in the prime factorization of  , no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor   of  , the prime   does not evenly divide  . Also   is square-free if and only if in every factorization  , the factors   and   are coprime. An immediate result of this definition is that all prime numbers are square-free.

A positive integer   is square-free if and only if all abelian groups of order   are isomorphic, which is the case if and only if any such group is cyclic. This follows from the classification of finitely generated abelian groups.

A integer   is square-free if and only if the factor ring   (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form   is a field if and only if   is prime.

For every positive integer  , the set of all positive divisors of   becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if   is square-free.

A positive integer   is square-free if and only if  , where   denotes the Möbius function.

Dirichlet seriesEdit

The absolute value of the Möbius function is the indicator function for the square-free integers – that is, |μ(n)| is equal to 1 if n is square-free, and 0 if it is not. The Dirichlet series of this indicator function is


where ζ(s) is the Riemann zeta function. This follows from the Euler product


where the products are taken over the prime numbers.


Let Q(x) denote the number of square-free integers between 1 and x (OEISA013928 shifting index by 1). For large n, 3/4 of the positive integers less than n are not divisible by 4, 8/9 of these numbers are not divisible by 9, and so on. Because these ratios satisfy the multiplicative property (this follows from Chinese remainder theorem), we obtain the approximation:


This argument can be made rigorous for getting the estimate (using big O notation)


Sketch of a proof: the above characterization gives


observing that the last summand is zero for  , it results that


By exploiting the largest known zero-free region of the Riemann zeta function Arnold Walfisz improved the approximation to[4]


for some positive constant c.

Under the Riemann hypothesis, the error term can be reduced to[5]


Recently (2015) the error term has been further reduced to[6]


The asymptotic/natural density of square-free numbers is therefore


Therefore over 3/5 of the integers are square-free.

Likewise, if Q(x,n) denotes the number of n-free integers (e.g. 3-free integers being cube-free integers) between 1 and x, one can show[7]


Since a multiple of 4 must have a square factor 4=22, it cannot occur that four consecutive integers are all square-free. On the other hand, there exist infinitely many integers n for which 4n +1, 4n +2, 4n +3 are all square-free. Otherwise, observing that 4n and at least one of 4n +1, 4n +2, 4n +3 among four could be non-square-free for sufficiently large n, half of all positive integers minus finitely many must be non-square-free and therefore

  for some constant C,

contrary to the above asymptotic estimate for  .

There exist sequences of consecutive non-square-free integers of arbitrary length. Indeed, if n satisfies a simultaneous congruence


for distinct primes  , then each   is divisible by pi 2.[8] On the other hand, the above-mentioned estimate   implies that, for some constant c, there always exists a square-free integer between x and   for positive x. Moreover, an elementary argument allows us to replace   by  [9] The ABC conjecture would allow  .[10]

Table of Q(x), 6/π2x, and R(x)Edit

The table shows how   and   compare at powers of 10.

  , also denoted as  .

10 7 6.1 0.9
102 61 60.8 0.2
103 608 607.9 0.1
104 6,083 6,079.3 3.7
105 60,794 60,792.7 1.3
106 607,926 607,927.1 - 1.3
107 6,079,291 6,079,271.0 20.0
108 60,792,694 60,792,710.2 - 16.2
109 607,927,124 607,927,101.9 22.1
1010 6,079,270,942 6,079,271,018.5 - 76.5
1011 60,792,710,280 60,792,710,185.4 94.6
1012 607,927,102,274 607,927,101,854.0 420.0
1013 6,079,271,018,294 6,079,271,018,540.3 - 246.3
1014 60,792,710,185,947 60,792,710,185,402.7 544.3
1015 607,927,101,854,103 607,927,101,854,027.0 76.0

  changes its sign infinitely often as   tends to infinity.[11]

The absolute value of   is astonishingly small compared with  .

Encoding as binary numbersEdit

If we represent a square-free number as the infinite product


then we may take those   and use them as bits in a binary number with the encoding


The square-free number 42 has factorization 2 × 3 × 7, or as an infinite product 21 · 31 · 50 · 71 · 110 · 130 ··· Thus the number 42 may be encoded as the binary sequence ...001011 or 11 decimal. (The binary digits are reversed from the ordering in the infinite product.)

Since the prime factorization of every number is unique, so also is every binary encoding of the square-free integers.

The converse is also true. Since every positive integer has a unique binary representation it is possible to reverse this encoding so that they may be decoded into a unique square-free integer.

Again, for example, if we begin with the number 42, this time as simply a positive integer, we have its binary representation 101010. This decodes to 20 · 31 · 50 · 71 · 110 · 131 = 3 × 7 × 13 = 273.

Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree integers.

(See sequences A019565, A048672 and A064273 in the OEIS.)

Erdős squarefree conjectureEdit

The central binomial coefficient


is never squarefree for n > 4. This was proven in 1985 for all sufficiently large integers by András Sárközy,[12] and for all integers > 4 in 1996 by Olivier Ramaré and Andrew Granville.[13]

Squarefree coreEdit

Let us call "t-free" a positive integer that has no t-th power in its divisors. In particular, the 2-free integers are the square-free integers.

The multiplicative function   maps every positive integer n to the quotient of n by its largest divisor that is a t-th power. That is,


The integer   is t-free, and every t-free integer is mapped to itself by the function  

The Dirichlet generating function of the sequence   is


See also OEISA007913 (t=2), OEISA050985 (t=3) and OEISA053165 (t=4).


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  2. ^ Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (1 September 2004). "PRIMES is in P" (PDF). Annals of Mathematics. 160 (2): 781–793. doi:10.4007/annals.2004.160.781. ISSN 0003-486X. MR 2123939. Zbl 1071.11070.
  3. ^ Richards, Chelsea (2009). Algorithms for factoring square-free polynomials over finite fields (PDF) (Master's thesis). Canada: Simon Fraser University.
  4. ^ Walfisz, A. (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften.
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  7. ^ Linfoot, E. H.; Evelyn, C. J. A. (1929). "On a Problem in the Additive Theory of Numbers". Mathematische Zeitschrift. 30: 443–448. doi:10.1007/BF01187781. S2CID 120604049.
  8. ^ Parent, D. P. (1984). Exercises in Number Theory. Problem Books in Mathematics. Springer-Verlag New York. doi:10.1007/978-1-4757-5194-9. ISBN 978-1-4757-5194-9.
  9. ^ Filaseta, Michael; Trifonov, Ognian (1992). "On gaps between squarefree numbers. II". Journal of the London Mathematical Society. Second Series. 45 (2): 215–221. doi:10.1112/jlms/s2-45.2.215. MR 1171549.
  10. ^ Andrew, Granville (1998). "ABC allows us to count squarefrees". Int. Math. Res. Not. 1998 (19): 991–1009. doi:10.1155/S1073792898000592.
  11. ^ Minoru, Tanaka (1979). "Experiments concerning the distribution of squarefree numbers". Proceedings of the Japan Academy, Series A, Mathematical Sciences. 55 (3). doi:10.3792/pjaa.55.101. S2CID 121862978.
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