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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 greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor p k, where prime and k is odd. In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are ...
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about geometrical figures; or rather to teach them not to reason using figures, but to use only textual descriptions and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations.
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 ...
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 .
Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics includes the aesthetics and culture of mathematics, peculiar or amusing stories and coincidences about mathematics, and the personal lives of mathematicians.
The square root of a quantity strongly related to the discriminant appears in the formulas for the roots of a cubic polynomial. Specifically, this quantity can be −3 times the discriminant, or its product with the square of a rational number; for example, the square of 1/18 in the case of Cardano formula.
More generally, for any square-free integer , the quadratic field is a number field obtained by adjoining the square root of to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, d = − 1 {\displaystyle d=-1} .