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Some unconventional problems in number theory. (English) Zbl 0399.10001
Astérisque 61, 73-82 (1979).
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 0032.01301)] 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 \(d_1< d_2< d_1[1+(\varepsilon/3)^{1-\eta\log\log n}]\), in spite of a previous claim that this had been proved. As another example, if \(p_1^{(n)}< \dots< p_{v(n)}^{(n)}\) 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 \(\varepsilon > 0\), \(\eta > 0\) there is a \(c=c(\varepsilon,\eta)\) such that the density is greater than \(1-\eta\) for the set of integers \(n\) for which every \(c < v\leq v(n)\), \(v(1-\varepsilon) < \log\log p_v^{(n)} < (1+\varepsilon)v\). 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 \(1/2\)? 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.
For the entire collection see [Zbl 0394.00007].
Reviewer: P.Garrison

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11N05 Distribution of primes
11B83 Special sequences and polynomials
00A07 Problem books