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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 ...
To factorize the integer n, Fermat's method entails a search for a single number a, n 1/2 < a < n−1, such that the remainder of a 2 divided by n is a square. But these a are hard to find. The quadratic sieve consists of computing the remainder of a 2 /n for several a, then finding a subset of these whose product is a square. This will yield a ...
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.
Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
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.
A simple example is the Fermat factorization method, which considers the sequence of numbers :=, for := ⌈ ⌉ +. If one of the x i {\displaystyle x_{i}} equals a perfect square b 2 {\displaystyle b^{2}} , then N = a i 2 − b 2 = ( a i + b ) ( a i − b ) {\displaystyle N=a_{i}^{2}-b^{2}=(a_{i}+b)(a_{i}-b)} is a (potentially non-trivial ...