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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.
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.
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 .
In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large.
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There is no known polynomial-time algorithm for computing the square-free part of an integer. [ 3 ] The definition is generalized to the largest t {\displaystyle t} -free divisor of n {\displaystyle n} , r a d t {\displaystyle \mathrm {rad} _{t}} , which are multiplicative functions which act on prime powers as