On the largest prime factors of consecutive integers in short intervals

Abstract:

For an integer $n>1$, let $P(n)$ be the largest prime factor of $n$. We prove that, for $x\rightarrow \infty$, there exists a positive proportion of consecutive integers $n$ and $n+1$ such that $P(n)<P(n+1)$ in short intervals $(x, x+y]$ with $x^{7/12}<y\leqslant x.$ In particular, we have \[ \big |\{n\leqslant x: P(n)< P(n+1)\}\big |> 0.1063 x. \] This improves a previous result of La Bretèche, Pomerance and Tenenbaum.