Publications of (and about) Paul Erdös
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 d1 and d2 satisfying d1 < d2 < 2d1 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 d1 < d2 < d1[1+(\epsilon/3)1-\eta log log n], in spite of a previous claim that this had been proved. As another example, if p1(n) < ... < pv(n)(n) are the consecutive prime factors of n, then for almost all n the v-th prime factor of n satisfies log log pv(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 pv(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.
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
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