On the largest prime factors of consecutive integers

Denote by \(P^+(n)\) $P^{+}(n)$ the largest prime factor of the integer n. One of Erdős and Turán’s conjectures asserts that the asymptotic density of integers n satisfying \(P^+(n)<P^+(n+1)\) $P^{+}(n)<P^{+}(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.2017, which improves the previous result “0.1356” of the second author.