Publications of Paul Erdos

Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No: 151.03502
Autor: Erdös, Pál; Sarközy, A.; Szemeredi, E.
Title: On the solvability of the equations [a_i,a_j] = a_r and (a'_i,a'_j) = a'_r in sequences of positive density (In English)
Source: J. Math. Anal. Appl. 15, 60-64 (1966).
Review: The authors obtain the following results.
1) Let a₁ < a₂ < ··· be an infinite sequence of integers for which there are infinitely many integers n₁ < n₂ < ··· satisfying

sum_{a_i < n_k} {1 \over a_i} > c₁ {log n_k \over (log log n_k)^½}.

Then the equations (a'_i,a'_j) = a'_r,[a_i,a_j] = a_r have infinitely many solutions. The symbol (a_i,a_j) denotes the greatest common divisor and [a_i,a_j] denotes the least common multiple of a_i and a_j.
2) Let a₁ < a₂ < ··· be an infinite sequence of integers for which there are infinitely many integers n₁ < n₂ < ··· satisfying

sum_{a_i < n_k} {1 \over a_i} > c₂ {log n_k \over (log log n_k)^1/4}.

Then there are infinitely many quadruplets of distincts integers a_i,a_j,a_r,a_s satisfying (a_i,a_j) = a_r, [a_i,a_j] = a_s, c₁ and c₂ denote suitable positive constants.
Reviewer: Cs.Pogany
Classif.: * 11B83 Special sequences of integers and polynomials
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