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A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
In mathematics, a sum of radicals is defined as a finite linear combination of n th roots: =, where , are natural numbers and , are real numbers.. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
The main difficulty is that, in order to solve the problem, the square-roots should be computed to a high accuracy, which may require a large number of bits. The problem is mentioned in the Open Problems Garden. [4] Blomer [5] presents a polynomial-time Monte Carlo algorithm for deciding whether a
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. This equation was first studied extensively in India starting with Brahmagupta , [ 1 ] who found an integer solution to 92 x 2 + 1 = y 2 {\displaystyle 92x^{2}+1=y^{2}} in his Brāhmasphuṭasiddhānta circa 628. [ 2 ]
Newton's method is a powerful technique—if the derivative of the function at the root is nonzero, then the convergence is at least quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties ...
Tonelli–Shanks cannot be used for composite moduli: finding square roots modulo composite numbers is a computational problem equivalent to integer factorization. [ 1 ] An equivalent, but slightly more redundant version of this algorithm was developed by Alberto Tonelli [ 2 ] [ 3 ] in 1891.
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