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**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 d_{1} < d_{2} < 2d_{1}, so that all prime factors of d_{1}d_{2} are > k is less than \epsilon^{2}? (2) Is there a covering system q_{i}(mod n)_{i}, 1 < n_{1} < ... < n_{k} for which n_{1} 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/a_{i} = oo then the integers a_{i} contain, for every k, k consecutive terms of an arithmetic progression. (6) Let 1 \leq a_{1} \leq ... \leq a_{tn} \leq n, where (a_{i}+a_{j})\nmid a_{i}a_{j}. What is the behavior of **max** t_{n}? (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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