Problem Statement
Let $\alpha,\beta\in (0,1)$ and let $P(n)$ denote the largest prime divisor of $n$. Does the density of integers $n$ such that $P(n)<n^{\alpha}$ and $P(n+1)<(n+1)^\beta$ exist?
Categories:
Number Theory
Progress
Dickman [Di30] showed the density of smooth $n$, with largest prime factor $<n^\alpha$, is $\rho(1/\alpha)$ where $\rho$ is the Dickman function.Erdős also asked whether infinitely many such $n$ even exist, but Meza has observed that this follows immediately from Schinzel's result [Sc67b] that for infinitely many $n$ the largest prime factor of $n(n+1)$ is at most $n^{O(1/\log\log n)}$.
Erdős asked whether the events $P(n)<n^\alpha$ and $P(n+1)<(n+1)^\beta$ are independent, in the sense that the density of $n$ satisfying both conditions is equal to $\rho(1/\alpha)\rho(1/\beta)$.
Teräväinen [Te18] has proved the logarithmic density exists and is equal to $\rho(1/\alpha)\rho(1/\beta)$.
Wang [Wa21] has proved the density is $\rho(1/\alpha)\rho(1/\beta)$ assuming the Elliott-Halberstam conjecture for friable integers.
See also [370].
Source: erdosproblems.com/928 | Last verified: January 19, 2026