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In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
For univariate polynomials over the rationals (or more generally over a field of characteristic zero), Yun's algorithm exploits this to efficiently factorize the polynomial into square-free factors, that is, factors that are not a multiple of a square, performing a sequence of GCD computations starting with gcd(f(x), f '(x)). To factorize the ...
The polynomial P = x 4 + 1 is irreducible over Q but not over any finite field. On any field extension of F 2, P = (x + 1) 4. On every other finite field, at least one of −1, 2 and −2 is a square, because the product of two non-squares is a square and so we have; If =, then = (+) ().
As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where 13729 × 1372933 = 18848997157, Fermat's factorization method will begin with ⌈ √ n ⌉ = 18848997159 which immediately yields b = √ a 2 − n = √ 4 = 2 and hence the factors a − b = 18848997157 and a + b = 18848997161.
The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.
Now the product of the factors a − mb mod n can be obtained as a square in two ways—one for each homomorphism. Thus, one can find two numbers x and y, with x 2 − y 2 divisible by n and again with probability at least one half we get a factor of n by finding the greatest common divisor of n and x − y.
Similarly, a 2×2×3 experiment has three factors, two at 2 levels and one at 3, for a total of 12 treatment combinations. If every factor has s levels (a so-called fixed-level or symmetric design), the experiment is typically denoted by s k, where k is the number of factors. Thus a 2 5 experiment has 5 factors
The quadratic sieve attempts to find pairs of integers x and y(x) (where y(x) is a function of x) satisfying a much weaker condition than x 2 ≡ y 2 (mod n). It selects a set of primes called the factor base, and attempts to find x such that the least absolute remainder of y(x) = x 2 mod n factorizes completely over