Analysis of a Simple Factorization Algorithm

The probability that the k-th largest prime factor of a number n is at most n to the x power is shown to approach a limit F sub Kx as n approaches infinity. Several interesting properties of F sub Kx are explored, and numerical tables are given. These results are applied to the analysis of an algorithm commonly used to find all prime factors of a given number. The average number of digits in the k-th largest prime factor of a random m-digit number is shown to be asymptotically equivalent to the average length of the k-th longest cycle in a permutation on m objects.