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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 ...
Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).
A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
A general-purpose factoring algorithm, also known as a Category 2, Second Category, or Kraitchik family algorithm, [10] has a running time which depends solely on the size of the integer to be factored. This is the type of algorithm used to factor RSA numbers. Most general-purpose factoring algorithms are based on the congruence of squares method.
Its exponent in the decomposition, 2, is even. Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2. The prime decomposition of the number 3430 is 2 · 5 · 7 3. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum of two squares.
It follows that arbitrarily large prime numbers can be found as the prime factors of the numbers !, leading to a proof of Euclid's theorem that the number of primes is infinite. [35] When n ! ± 1 {\displaystyle n!\pm 1} is itself prime it is called a factorial prime ; [ 36 ] relatedly, Brocard's problem , also posed by Srinivasa Ramanujan ...
RSA-140 has 140 decimal digits (463 bits), and was factored on February 2, 1999, by a team led by Herman te Riele and composed of Stefania Cavallar, Bruce Dodson, Arjen K. Lenstra, Paul Leyland, Walter Lioen, Peter L. Montgomery, Brian Murphy and Paul Zimmermann. [12] [13] The value and factorization are as follows:
The square-free number 42 has factorization 2 × 3 × 7, or as an infinite product 2 1 · 3 1 · 5 0 · 7 1 · 11 0 · 13 0 ··· Thus the number 42 may be encoded as the binary sequence ...001011 or 11 decimal. (The binary digits are reversed from the ordering in the infinite product.)