On rough and smooth neighbors (Q2472510)

For every integer \(n\geq 2\), let \(P^{+}(n)\) and \(P^{-}(n)\) denote the largest and the smallest prime factors of \(n\). An integer \(n\) is said to be \(y\)-smooth if \(P^{+}(n)\leq y\). In this paper the authors consider the quotients \({\mathcal F}(n)=P^{+}(n)/P^{-}(n+1)\) and \({\mathcal G}(n)=P^{+}(n+1)/P^{-}(n)\). The authors show that although these quantities tend to be large, the value sets are quite dense in the set of all positive real numbers. In particular, both value sets contain all fractions of the form \(p/q>1\) and almost all fractions of the form \(p/q<1\), where \(p\) and \(q\) are primes. On the other hand, it is shown that for every prime \(p\), there are infinitely many primes \(q\) such that \(p/q\) is not a value of \({\mathcal F}(n)\). The arguments use some results from the theory of \(y\)-smooth integers (`psixyology') and analytic number theory. Some open problems are formulated.