Erdos Problem #371: do the integers with P(n) < P(n+1) have density 1/2? (Erdos-Pomerance joint independence of largest prime factors)

Theorem 1.1: for fixed $a,b\in(0,1)$, $\frac1X\#\{2\le n\le X:P^+(n)\le n^a,P^+(n+1)\le n^b\}\to\rho(1/a)\rho(1/b)$, unconditionally and at every large scale. Corollary 1.2: $P^+(n)<P^+(n+1)$ and the reverse ordering each have natural density $1/2$. Inclusion-exclusion also gives the Erdos-Pomerance upper-tail independence statement. It gives no rate and no uniformity in $a,b$ (unlike the almost-all-scales quantitative law of Tao-Teravainen 2026), and says nothing about three consecutive integers (Erdos #372 and Balog's density 1/6 conjecture).

Theorem 1.1: for fixed $a,b\in(0,1)$, $\frac1X\#\{2\le n\le X:P^+(n)\le n^a,P^+(n+1)\le n^b\}\to\rho(1/a)\rho(1/b)$, unconditionally and at every large scale. Corollary 1.2: $P^+(n)<P^+(n+1)$ and the reverse ordering each have natural density $1/2$. Inclusion-exclusion also gives the Erdos-Pomerance upper-tail independence statement. It gives no rate and no uniformity in $a,b$ (unlike the almost-all-scales quantitative law of Tao-Teravainen 2026), and says nothing about three consecutive integers (Erdos #372 and Balog's density 1/6 conjecture).