Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  083.03702
Autor:  Erdös, Paul
Title:  On an elementary problem in number theory. (In English)
Source:  Can. Math. Bull. 1, 5-8 (1958).
Review:  Given 0 < x \leq y, the author seeks an estimate of the smallest f(x) so that there exist integers u,v satisfying (1) 0 \leq u, v < f(x) and (x+u,y+v) = 1. He proves that for every \epsilon > 0 there exists arbitrarily large x satisfying

f(x) > (1-\epsilon) (log x/ log log x) ½,    (2)

but for some c > 0 and all x, (3) f(x) < c log x/ log log x. The author indicates that it seems a difficult problem to get a sharp estimate of f(x). He proves also the Theorem. Let g(x) (log x/ log log x)-1 ––> oo, 0 \leq x < y. Then the number of pairs 0 \leq u, v < g(x) satisfying (x+u,y+v) = 1 equals (1+o (1))6 \pi-2 g2 (x).
Reviewer:  J.P.Tull
Classif.:  * 11N56 Rate of growth of arithmetic functions
Index Words:  number theory

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