On the largest prime factors of consecutive integers in short intervals

Source abstract

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