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If the approximate ratio of two factors (/) is known, then a rational number / can be picked near that value. N u v = c v ⋅ d u {\displaystyle Nuv=cv\cdot du} , and Fermat's method, applied to Nuv , will find the factors c v {\displaystyle cv} and d u {\displaystyle du} quickly.
Start with division by 2: the number is even, and n = 2 · 693. Continue with 693, and 2 as a first divisor candidate. 693 is odd (2 is not a divisor), but is a multiple of 3: one has 693 = 3 · 231 and n = 2 · 3 · 231. Continue with 231, and 3 as a first divisor candidate. 231 is also a multiple of 3: one has 231 = 3 · 77, and thus n = 2 ...
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4).
When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n 1/2. The size of these values is exponential in the size of n (see below). The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the ...
Cunningham numbers of the form b n − 1 can only be prime if b = 2 and n is prime, assuming that n ≥ 2; these are the Mersenne numbers. Numbers of the form b n + 1 can only be prime if b is even and n is a power of 2, again assuming n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2. Any factor of a Fermat ...
The great disadvantage of Euler's factorization method is that it cannot be applied to factoring an integer with any prime factor of the form 4k + 3 occurring to an odd power in its prime factorization, as such a number can never be the sum of two squares. Even odd composite numbers of the form 4k + 1 are often the product of two primes of the ...
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Wheel factorization with n = 2 × 3 × 5 = 30.No primes will occur in the yellow areas. Wheel factorization is a method for generating a sequence of natural numbers by repeated additions, as determined by a number of the first few primes, so that the generated numbers are coprime with these primes, by construction.