smallfactor

Smallfactor is a term used in number theory and computational mathematics to refer to the smallest nontrivial factor of a positive integer, equivalently its smallest prime factor. In many contexts, smallfactor serves as a preprocessing step in integer factorization, where extracting small prime factors can simplify subsequent algorithms such as elliptic curve factorization (ECM) or Pollard's rho.

Computing the smallfactor of a number n typically begins with a quick check for small primes, often via trial division by a precomputed list of primes up to a chosen bound. If none divide n, the smallfactor is either a larger prime factor of n or n itself if n is prime. Efficient implementations may employ wheel factorization or sieve-based methods to accelerate the check for small primes.

In practice, removing small factors can dramatically reduce the difficulty of factorization, since many integers have small prime divisors. For example, smallfactor(1001) = 7, since 1001 = 7 × 11 × 13, and smallfactor(29) = 29 since 29 is prime. In cryptographic and mathematical contexts, identifying and removing small factors is a standard step before applying more expensive algorithms for full factorization, such as ECM or the general number field sieve.

Variations and related concepts include bounded trial division, where factors up to a fixed bound B are sought, and the use of small-factor sieves in fast primality testing and factoring software. See also prime factorization, sieve of Eratosthenes, ECM, and Pollard's rho.