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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 ...
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.
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 ...
Dixon's factorization method; E. Euler's factorization method; F. Factor base; Fast Library for Number Theory; Fermat's factorization method; G. General number field ...
It was while researching perfect numbers that he discovered Fermat's little theorem. He invented a factorization method—Fermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4.
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This lack of unique factorization is a major difficulty for solving Diophantine equations. For example, many wrong proofs of Fermat's Last Theorem (probably including Fermat's "truly marvelous proof of this, which this margin is too narrow to contain") were based on the implicit supposition of unique factorization.