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
                   11P32 Additive questions involving primes
                   11N05 Distribution of primes
                   05A05 Combinatorial choice problems
                   00A07 Problem books
Keywords:  covering congruences; problems; sum of prime and powers of two
Citations:  Zbl.444.00010

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