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Gaps between integers with the same prime factors. (English) Zbl 0929.11031

Dressler conjectured that there is a prime between any two positive integers \(a<c\) with the same prime factors. The authors show herein that if the \(abc\)-conjecture is true then \(c-a> a^{1/2-o(1)}\). Moreover it is believed that the maximal gap between consecutive primes \(\leq x\) is \(O(\log^2x)\). If both conjectures hold then Dressler’s conjecture holds for sufficiently large \(a<c\). The authors note that there are many examples of \(c-a<a^{1/2}\) but conjecture that \(c-a>a^{1/3}\) always. They prove \(c-a\gg_\varepsilon (\log a)^{3/4-\varepsilon}\). They also carefully consider the case where \(a\) and \(c\) have only two prime factors.

MSC:

11N05 Distribution of primes
11N25 Distribution of integers with specified multiplicative constraints
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