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The multiplicity of a prime factor p of n is the largest exponent m for which p m divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p 1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
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 ...
As each iteration is greater than the previous up until a prime is reached, factorizations generally grow more difficult so long as an end is not reached. As of August 2016 [update] the pursuit of HP(49) concerns the factorization of a 251-digit composite factor of HP49(119) after a break was achieved on 3 December 2014 with the calculation of ...
For prime powers, efficient classical factorization algorithms exist, [22] hence the rest of the quantum algorithm may assume that is not a prime power. If those easy cases do not produce a nontrivial factor of N {\displaystyle N} , the algorithm proceeds to handle the remaining case.
A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes (250s BCE), the sieve of Sundaram (1934), the still faster but more complicated sieve of Atkin [1] (2003), sieve of Pritchard (1979), and various wheel sieves [2] are most common.
The factorizations take the form of an optional unit multiplied by integer powers of Gaussian primes. Note that there are rational primes which are not Gaussian primes. A simple example is the rational prime 5, which is factored as 5=(2+i)(2−i) in the table, and therefore not a Gaussian prime.
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. [1] It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized.
For computing the factorization of an integer n, one needs an algorithm for finding a divisor q of n or deciding that n is prime. When such a divisor is found, the repeated application of this algorithm to the factors q and n / q gives eventually the complete factorization of n .