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