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On gaps between \(k\)-free numbers. (English) Zbl 0841.11046

Let \(k\) be an integer \(\geq 2\) and let \(\theta_k= \inf\{ \theta\): there exists \(x(k)\) such that if \(x\geq x(k)\), then the interval \([x,x+ \theta]\) contains a \(k\)-free integer}. The problem of evaluating \(\theta_k\) was investigated by many authors. The best previously known results are due to M. Filaseta for \(k=2\) \((\theta_2\leq1/5)\) and the author for \(k\geq 3\) \((\theta_3\leq 7/46)\) and for \(k\geq 4\) \((\theta_3\leq 1/(2k+ 1/2)\)). The author improves this: Theorem. For every integer \(k\geq 3\) there exists a constant \(c(k)\) such that if \(x\) is sufficiently large then the interval \((x,x+ c(k) x^{1/(2k +1)} \log x]\) contains a \(k\)-free integer. The method of the proof was developed by M. Filaseta and the author and is elementary.

MSC:

11N25 Distribution of integers with specified multiplicative constraints
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