Publications of Paul Erdos

Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No: 159.06003
Autor: Erdös, Pál; Sarközi, A.; Szemeredi, E.
Title: On the solvability of some equations in dense sequences of integers (In English. RU original)
Source: Sov. Math., Dokl. 8, 1160-1164 (1967); translation from Dokl. Akad. Nauk SSSR 176, 541-544 (1967).
Review: Let a₁ < a₂ < ... be an infinite sequence of integers for which the equation (a_i,a_j) = a_r is unsolvable for any set of distinct indices i,j,r. The authors prove that then

sum_{k = 1}^oo {1 \over a_k log a_k} < oo. (1)

The proof is elementary but quite complicated and uses combinatorial arguments. In a previous paper [P. Erdös, J. London Math. Soc. 10, 126-128 (1935; Zbl 012.05202)] the following weaker result was proved: Assume that no a_i divides any other than (1) holds. In another paper [J. Math. Anal. Appl. 15, 60-64 (1966; Zbl 151.03502)] the authors point out that (1) does not hold if we assume that [a_i,a_j] = a_r is unsolvable for any set of distinct indices i,j,r.
Classif.: * 11B83 Special sequences of integers and polynomials
11B75 Combinatorial number theory
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