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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 399.10001

**Autor: ** Erdös, Paul

**Title: ** Some unconventional problems in number theory. (In English)

**Source: ** Asterisque 61, 73-82 (1979).

**Review: ** Many interesting problems and clusters of problems, solved and unsolved, are listed here. For example, the author has proved [Bull. Am. Math. Soc. 54, 685-692 (1948; Zbl 032.01301)] that the set of integers having two divisors d_{1} and d_{2} satisfying d_{1} < d_{2} < 2d_{1} does have a density, but it is still an open question as to whether that density is 1. It is also not yet known whether or not almost all integers n have two divisors satisfying d_{1} < d_{2} < d_{1}[1+(\epsilon/3)^{1-\eta log log n}], in spite of a previous claim that this had been proved. As another example, if p_{1}^{(n)} < ... < p_{v(n)}^{(n)} are the consecutive prime factors of n, then for almost all n the v-th prime factor of n satisfies log log p_{v}^{(n)} = (1+o(1))v or, more precisely, for every \epsilon > 0, \eta > 0 there is a c = c(\epsilon,\eta) such that the density is greater than 1-\eta for the set of integers n for which every c < v \leq v(n), v(1-\epsilon) < log log p_{v}^{(n)} < (1+\epsilon)v. This is the only result for which a proof is provided in this paper. A final example: Let P(n) be the greatest prime factor of n. Is it true that the density of the set of integers n satisfying P(n+1) > P(n) is ½? Is it true that the density of the set of integers n for which P(n+1) > P(n)n^{\alpha} exists for every \alpha? The author warns that this problem is probably very difficult.

**Reviewer: ** P.Garrison

**Classif.: ** * 11-02 Research monographs (number theory)

11N05 Distribution of primes

11B83 Special sequences of integers and polynomials

00A07 Problem books

**Keywords: ** greatest prime factor; divisor problems; consecutive prime factors; density

**Index Words: ** problems

**Citations: ** Zbl.032.013

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