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Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
Wheel factorization; Pollard's rho algorithm, which has two common flavors to identify group cycles: one by Floyd and one by Brent. Algebraic-group factorization algorithms, among which are Pollard's p − 1 algorithm, Williams' p + 1 algorithm, and Lenstra elliptic curve factorization; Fermat's factorization method; Euler's factorization method
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes.. A Gaussian integer is a complex number + such that a and b are integers. The norm (+) = + of a Gaussian integer is an integer equal to the square of the absolute value of the Gaussian integer.
Given a positive integer n, Fermat's factorization method relies on finding numbers x and y satisfying the equality = We can then factor n = x 2 − y 2 = (x + y)(x − y). This algorithm is slow in practice because we need to search many such numbers, and only a few satisfy the equation.
In number theory, Dixon's factorization method (also Dixon's random squares method [1] or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures about the smoothness ...
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Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
Fermat (named after Pierre de Fermat) is a program developed by Prof. Robert H. Lewis of Fordham University.It is a computer algebra system, in which items being computed can be integers (of arbitrary size), rational numbers, real numbers, complex numbers, modular numbers, finite field elements, multivariable polynomials, rational functions, or polynomials modulo other polynomials.