Integers without large prime factors

Abstract

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Roughly squarefree values of the Euler and Carmichael functions

“…To prove this, let P = P (n) be the largest prime factor of one such n, and write n = P m. Using well known results about the distribution of smooth numbers (see, for example, [6]), we have…”

An estimate for the number of integers without large prime factors

“…The above E(m) provides an approximation of ξ u , where ξ u is the unique solution to the equation e ξu = 1 + uξ u , and E(m) can be used in some other estimate formulas for Ψ(x, y) (for example, see Section 2 of [9]). We prove that the error term of our estimate for α u is O(1/(u(log y) 2 )) for x → ∞, in the range…”

“…The behavior of Ψ(x, y) has been investigated by many authors ( [2], [3], [4], [5], [6], [7], [8], [10], [13]). We see good summaries for the investigations of Ψ(x, y) in [9] and [11].…”

Primeless Factoring-Based Cryptography

“…N , where P r (x, N) = #{1 ≤ n ≤ N|n r ≤ N x }; -ψ(x, y) = P 1 log y log x , x is the de Brujin function [10] and the ratio ψ(x, y)/x can be interpreted as the probability that an integer chosen at random in the interval [1, x] has all its prime factors smaller than or equal to y. This function has several approximations [10,13].…”