On the largest prime factors of consecutive square-free integers

For an integer n>1, let P+(n) be the largest prime factor of n. Following a celebrated conjecture of Erdős and Turán in the 1930s, Erdős and Pomerance proved in 1978 that lim infx→∞|{n≤x:P+(n+1)>P+(n)}|x>0. In this article, their result is extended to lim infx→∞|{n≤x:P+(n+1)>P+(n),μ2(n)=μ2(n+1)=1}|x>0.