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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.
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. [1] [2] A more general definition includes all positive rational numbers with this property. [3] The sequence of (integer) congruent numbers starts with
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 .
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.
For a nonzero square free integer , the discriminant of the quadratic field = is if is congruent to modulo , and otherwise . For example, if d {\displaystyle d} is − 1 {\displaystyle -1} , then K {\displaystyle K} is the field of Gaussian rationals and the discriminant is − 4 {\displaystyle -4} .
An integer is a fundamental discriminant if and only if it meets one of the following criteria: Case 1: is congruent to 1 modulo 4 (()) and is square-free, meaning it is not divisible by the square of any prime number.
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