Authors : Farida F. Sharifullina and Shamil T. Ishmukhametov
Abstract: A positive integer n is called the y-smooth for some positive number y if all prime divisors n are bounded above by the number y. A natural number n is called the y-power smooth if every prime power dividing n is bounded above by the number y. In order to assess the cryptosecurity of some algorithms with public-key encryption such as the known method RSA, it is necessary to be able to calculate the function concerning the number of smooth and power smooth numbers within the set numeric intervals. Each y power smooth integer n is also an y-smooth but the reverse is not true. Let’s denote by ψ(x, y) the amount of y-smooth integers in the range from 0 to x and by ψ*(x, y) the amount of y-power smooth numbers ranging from 0 to x. The definition implies that ψ*(x, y)≤ψ(x, y). In the scientific literature, one may meet a large number of publications devoted to the algorithm for calculating or approximating the function ψ(x, y). However, the publications by the function ψ*(x, y) are almost absent. At large x and y the values of these functions are similar, however for large x and small y the values of ψ(x, y) and ψ*(x, y) are significantly different. In this study, we give an overview of algorithms for calculating the amount of smooth and power smooth numbers at predetermined intervals and show our own results and observations.
Farida F. Sharifullina and Shamil T. Ishmukhametov, 2015. About Power Smooth Numbers. Research Journal of Applied Sciences, 10: 381-384.