Smooth Number - Powersmooth Numbers

Powersmooth Numbers

Further, m is called B-powersmooth if all prime powers dividing m satisfy:

For example, 243251 is 5-smooth, but is not 5-powersmooth. It is 16-powersmooth since its greatest prime factor power is 24 = 16. The number is also 17-powersmooth, 18-powersmooth, etc.

B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-smooth for some unspecified small number B; as B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O(B1/2) for groups of B-smooth order.