Publications of Paul Erdos

Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No: 151.03501
Autor: Erdös, Pál
Title: On some properties of prime factors of integers (In English)
Source: Nagoya Math. J. 27, 617-623 (1966).
Review: Let n = prod_{i = 1}^{\nu (n)} be the canonical decomposition of an integer n > 1. Define for 2 \leq j \leq \nu(n)

prod_{i = 1}^j-1 p_i^{\alpha i} = p_j^\gamma_j(n)

and set

max_{2 \leq j \leq \nu(n)} \gamma_j(n) = P(n).

The author proves the following results:
(1) for almost all integers n (i. e. for all integers n but possibly a sequence of integers of density 0) one has

P(n) = (1+o(1)) log₃ n / log₄ n;

(2) there is a continuous strictly increasing function \phi (c) with \phi (0) = 0, \phi (oo) = 1 such that for almost all integers n

{1 \over log₂n} sum_{\gamma_j (n) \leq c} 1 ––> \phi (c);

(3) the density of integers n for which max_{2 \leq j \leq \nu (n)} \gamma_i(n) < c/ log₂ n is given by \psi (c), where \psi (c) is a continuuous strictly increasing function with \psi (0) = 0, \psi(oo) = 1. Here, log₁ n = log n and log_kn = log (log_k-1 n) for k = 2,3,4.
Reviewer: S.Uchiyama
Classif.: * 11N25 Distribution of integers with specified multiplicative constraints
Index Words: number theory

Books Problems Set Theory Combinatorics Extremal Probl/Ramsey Th.

Graph Theory Add.Number Theory Mult.Number Theory Analysis Geometry

Probabability Personalia About Paul Erdös Publication Year Home Page

Books	Problems	Set Theory	Combinatorics	Extremal Probl/Ramsey Th.
Graph Theory	Add.Number Theory	Mult.Number Theory	Analysis	Geometry
Probabability	Personalia	About Paul Erdös	Publication Year	Home Page