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The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. 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.
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 ...
Weber showed that these fields have odd class number. In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 10 7, [9] and later improved this bound to 10 9. [10] These fields are the n-th layers of the cyclotomic Z 2-extension of Q.
This factorization is also unique up to the choice of a sign. For example, + + + = + + + is a factorization into content and primitive part. Gauss proved that the product of two primitive polynomials is also primitive (Gauss's lemma). This implies that a primitive polynomial is irreducible over the rationals if and only if it is irreducible ...
This representation is called the canonical representation [10] of n, or the standard form [11] [12] of n. For example, 999 = 3 3 ×37, 1000 = 2 3 ×5 3, 1001 = 7×11×13. Factors p 0 = 1 may be inserted without changing the value of n (for example, 1000 = 2 3 ×3 0 ×5 3).
Because the prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number. [8] Due to their ease of use in calculations involving fractions , many of these numbers are used in traditional systems of measurement and engineering designs.