Square free integer

Square-free factors of integers

The radical of an integer is its largest square-free factor. An integer is square-free if and only if it is equal to its radical.

Any arbitrary positive integer n can be represented in a unique way as the product of a powerful number and a square-free integer, which are coprime. The square-free factor is the largest square-free divisor k of n that is coprime with n/k.

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

In this factorization, m is the largest divisor of n such that m² is a divisor of n.

Equivalent characterizations

Every prime number is square-free.

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

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

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

For every positive integer n, the set of all positive divisors of n 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 n is square-free.

A positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.

A positive integer n is squarefree if and only if

This results from the properties of Möbius function, and the fact that this sum is equal to ∑ d ∣ m μ ( d ) , where m is the largest divisor of n such that m² divides n.

Dirichlet generating function

The Dirichlet generating function for the square-free numbers is

This is easily seen from the Euler product

Distribution

Let Q(x) denote the number of square-free integers between 1 and x. 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 events are independent, we obtain the approximation:

This argument can be made rigorous, and a very elementary estimate yields

(see pi and big O notation) since we use the above characterization to obtain

and, observing that the last summand is zero for d > x , we have

By exploiting the largest known zero-free region of the Riemann zeta function, due to Ivan Matveyevich Vinogradov, N.M. Korobov and Hans-Egon Richert, the maximal size of the error term has been reduced by Arnold Walfisz and we have

for some positive constant c. Under the Riemann hypothesis, the error term can be further reduced to yield

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

where ζ is the Riemann zeta function and 1/ζ(2) is approximately 0.6079. 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

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

contrary to the above asymptotic estimate for Q ( x ) .

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

for distinct primes p 1 , p 2 , … , p l , then each n + i is divisible by p_i ². On the other hand, the above-mentioned estimate Q ( x ) = 6 x / π 2 + O ( x ) implies that, for some constant c, there always exists a square-free integer between x and x + c x for positive x. Moreover, an elementary argument allows us to replace x + c x by x + c x 1 / 5 log ⁡ x . The ABC conjecture would allow x + x o ( 1 ) .

Encoding as binary numbers

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

then we may take those a n 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. (Note that 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 the in-order encodings of the square-free integers are a permutation of the set of all integers.

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

Erdős squarefree conjecture

The central binomial coefficient

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

Squarefree core

The multiplicative function c o r e t ( n ) is defined to map positive integers n to t-free numbers by reducing the exponents in the prime power representation modulo t:

The value set of c o r e 2 , in particular, are the square-free integers. Their Dirichlet generating functions are

OEIS representatives are  A007913 (t=2),  A050985 (t=3) and  A053165 (t=4).