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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.
Square free numbers may be defined as products of primes that are all different. This definition is equivalent with the one that is given in the first sentence of the article, as 1 is the empty product of primes. Thus, presently, 1 is always defined to be square free. D.Lazard 08:30, 6 August 2017 (UTC) Thank you for your answer.
In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square. This means that every s such that s 2 ∣ r {\displaystyle s^{2}\mid r} is a unit of R .
A non-negative integer is a square number when its square root is again an integer. For example, =, so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free. For a non-negative integer n, the n th square number is n 2, with 0 2 = 0 being the zeroth one. The concept of square can be extended to some ...
However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci. [4] Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number. [5]
Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q(√ d) is a quadratic extension of Q. The class number of Q(√ d) is one if and only if the ring of integers of Q(√ d) is a principal ideal domain. The Baker–Heegner–Stark theorem can then be stated as follows:
The radical of any integer is the largest square-free divisor of and so also described as the square-free kernel of . [2] There is no known polynomial-time algorithm for computing the square-free part of an integer.
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