Erdős–Turán conjecture on largest prime factors of consecutive integers

Determine whether the asymptotic density of positive integers n satisfying P^+(n)<P^+(n+1) is equal to 1/2.

Background

The paper studies the relative sizes of the largest prime factors of consecutive integers, where P+(n) denotes the largest prime factor of n. It records the Erdős–Turán conjecture that the set of integers n for which P+(n)<P+(n+1) has asymptotic density 1/2.

The paper does not resolve this conjecture. Instead, its first application establishes a positive lower bound of 0.296 for the count of such integers up to x, improving earlier unconditional density bounds. Thus the conjectured density 1/2 remains an unresolved target beyond the result proved in the paper.

References

One of Erdős and Turán's conjectures (see [22]) asserts that the asymptotic density of integers $n$ satisfying $P+(n)<P+(n+1)$ is 1/2.

— Convolution-type Bombieri-Vinogradov theorem with well-factorable Weights, and its applications  (2608.13299 - Yang, 13 Aug 2026) in Section 1, Introduction, paragraph beginning “As for application, we consider the following questions.”