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 (ai,aj) = ar is unsolvable for any set of distinct indices i,j,r. The authors prove that then

sumk = 1oo {1 \over ak log ak} < 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 ai 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 [ai,aj] = ar 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

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