Square-free integer
different from one are square-free integers that are pairwise coprime.
The radical of an integer is its largest square-free factor, that is
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.
can be represented in a unique way as the product of a square and a square-free integer:
are distinct prime numbers, then the square-free part is
[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).
, no prime factor occurs with an exponent larger than one.
An immediate result of this definition is that all prime numbers are square-free.
This follows from the classification of finitely generated abelian groups.
This follows from the Chinese remainder theorem and the fact that a ring of the form
This partially ordered set is always a distributive lattice.
Let Q(x) denote the number of square-free integers between 1 and x (OEIS: A013928 shifting index by 1).
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] In 2015 the error term was further reduced (assuming also Riemann hypothesis) to[6] The asymptotic/natural density of square-free numbers is therefore Therefore over 3/5 of the integers are 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
There exist sequences of consecutive non-square-free integers of arbitrary length.
Indeed, if n satisfies a simultaneous congruence for distinct primes
implies that, for some constant c, there always exists a square-free integer between x and
(with the latter rounded to one decimal place) compare at powers of 10.
If we represent a square-free number as the infinite product then we may take those
(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.
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.
Thus binary encoding of squarefree numbers describes a bijection between the nonnegative integers and the set of positive squarefree integers.
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] Let us call "t-free" a positive integer that has no t-th power in its divisors.
maps every positive integer n to the quotient of n by its largest divisor that is a t-th power.
