For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
For every fixed 0<a,b<1, the natural density of integers n with P⁺(n)≤nᵃ and P⁺(n+1)≤nᵇ is ρ(1/a)ρ(1/b), where P⁺ is the largest prime factor and ρ is the Dickman function.
The reported natural-density result resolves the Erdős–Pomerance joint independence conjecture and implies that either ordering of the largest prime factors has density one half.
Can the joint Dickman law be proved in ordinary natural density?
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OpenAI reports independent largest prime factors of consecutive integers
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