smooth number

A k-smooth number n is an integer such that its prime divisorsPlanetmathPlanetmathPlanetmath are less than k. For example, 14824161510814728 is 7-smooth since its prime factorsMathworldPlanetmath are just 2 and 3 (and thus its divisorsMathworldPlanetmathPlanetmathPlanetmath are all either powers of those two primes or multiples of 6).

For small k, the number of smooth numbers less than a given x can be estimated with the formula


where π(x) is the prime counting function and pi is the ith prime.

Smooth numbers have many applications, such as the many classic factorization algorithms which specifically call for smooth numbers in their initialization steps, as well as fast Fourier transform algorithms that also require smooth numbers.

Title smooth number
Canonical name SmoothNumber
Date of creation 2013-03-22 13:20:25
Last modified on 2013-03-22 13:20:25
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05
Related topic RoughNumber