In mathematics, a square-free integer (or squarefree integer) is an integer that 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, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are

Square-free factorization

different from 1 are square-free integers that are pairwise coprime. This is called the square-free factorization of .

be the prime factorization are distinct prime numbers

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 integers

The square-free part can be represented in a unique way as the product of a largest possible square.

can be represented in a unique way as the product of a powerful number

The radical of an integer in the notation of the preceding section. The radical of an integer may be smaller than the square-free part; an integer is square-free if and only if it is equal to its radical.

the square-free part, and the largest square-free factor. Each is a factor of the next one. All are easily deduced from the prime factorization

The square-free part is , the square-free factor such that the quotient is a square is , and the largest square-free factor is .

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. In contrast, polynomial-time algorithms are known for primality testing.

Equivalent characterizations

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

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.

is square-free if and only if the factor ring (see modular arithmetic) is a product of fieldss. This follows from the Chinese remainder theorem is prime.

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 is square-free.

is square-free if and only if denotes the Möbius function.

Dirichlet series

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

where is the Riemann zeta function. This follows from the Euler product

where the products are taken over the prime numbers.

Distribution

Let Q(x) denote the number of square-free integers between 1 and x ( 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

, it follows that

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

for some positive constant .

Under the Riemann hypothesis, the error term can be reduced to

In 2015 the error term was further reduced (assuming also Riemann hypothesis) to

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

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

1. for some constant C,

.

There exist sequences of consecutive non-square-free integers of arbitrary length. Indeed, for every tuple of distinct primes, the Chinese remainder theorem guarantees the existence of an that satisfies the simultaneous congruence

Each is then divisible by . implies that, for some constant c, there always exists a square-free integer between x for positive x The abc conjecture.

Computation of Q(x)

The squarefree integers can be identified and counted in time by using a modified Sieve of Eratosthenes. If only is desired, and not a list of the numbers that it counts, then () can be used to compute in time. The largest known value of , for , was computed by Jakub Pawlewicz in 2011 using an algorithm that achieves time, and an algorithm taking time has been outlined but not implemented.

Table of Q(x), 6π2x, and R(x)

(with the latter rounded to one decimal place) compare at powers of 10.

.

1. {| class="wikitable" style="text-align: right"

|- | 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 | |- | 107 | 6,079,291 | 6,079,271.0 | 20.0 |- | 108 | 60,792,694 | 60,792,710.2 | |- | 109 | 607,927,124 | 607,927,101.9 | 22.1 |- | 1010 | 6,079,270,942 | 6,079,271,018.5 | |- | 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 | |- | 1014 | 60,792,710,185,947 | 60,792,710,185,402.7 | 544.3 |- | 1015 tends to infinity.

.

Encoding as binary numbers

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

and use them as bits in a binary number with the encoding

The square-free number 42 has factorization , or as an infinite product 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

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 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

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,

is t-free, and every t

The Dirichlet generating function is

1. .

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

Notes

References