Distribution of \({\mathcal B}\)-free numbers in short intervals
The author gives a brief survey of results on \(\mathcal B\)-free numbers, up to and including the result described in the preceding review, before considering the solubility of the equations \(p+ k= a\), \(n=p+ a\), where \(p\) denotes a prime and \(a\) a \(\mathcal B\)-free number. The new result is that these have solutions with \(p\) between \(n\) and \(n- n^ \theta\), whenever \(\theta>3/4\) and \(n\) is large enough (given a necessary condition mod 2 when \(n\) is even). The proof is supplied for the equation \(p+ k= a\), where (as in the author's previous theorem) the result embodies a quantitative lower bound of the correct order of magnitude, in this instance for the number of primes \(p\) involved.
scientific article; zbMATH DE number 484168