## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  448.10003
Autor:  Erdös, Paul
Title:  On some problems in number theory. (In English)
Source:  Theorie des nombres, Semin. Delange-Pisot-Poitou, Paris 1979-80, Prog. Math. 12, 71-75 (1981).
Review:  [For the entire collection see Zbl 444.00010.]
The problems discussed include or are related to the following conjectures: (1) Is it true that to every \epsilon > 0 there is a k so that the density of integers n for which n has two divisors d1 < d2 < 2d1, so that all prime factors of d1d2 are > k is less than \epsilon2? (2) Is there a covering system qi(mod n)i, 1 < n1 < ... < nk for which n1 is as large as desired? (3) Is it true that there is an r so that every integer is the sum of a prime and at most r powers of two? (4) There exists an F(n) such if the integers not exceeding F(n) are split into two classes at least one of them contains an arithmetic progression of n terms. Is lim F(n)1/n = oo? (5) If \Sigma 1/ai = oo then the integers ai contain, for every k, k consecutive terms of an arithmetic progression. (6) Let 1 \leq a1 \leq ... \leq atn \leq n, where (ai+aj)\nmid aiaj. What is the behavior of max tn? (7) Let F(X,n) be the number of integers X < m \leq X+n for which there is a p|m, n/3 < p < n/2. Is it true that F(X,n) > Cn/ log n?
Reviewer:  G.Lord
Classif.:  * 11-02 Research monographs (number theory)
11B83 Special sequences of integers and polynomials
11B25 Arithmetic progressions