Does the density of $n$ with $P(n)<n^\alpha$ and $P(n+1)<(n+1)^\beta$ exist? (Erdős #928)
Statement
Let $\alpha,\beta\in(0,1)$ and let $P(m)$ denote the largest prime divisor of $m$. Does the natural density of the set of integers $n$ satisfying both $$P(n)<n^{\alpha}\qquad\text{and}\qquad P(n+1)<(n+1)^{\beta}$$ exist? Erdős further asked whether the two smoothness events are asymptotically independent, i.e. whether this density equals $\rho(1/\alpha)\,\rho(1/\beta)$, where $\rho$ is the Dickman function.
Acceptance. FULLY RESOLVES (proof-shaped): a complete written or machine-checkable proof that the natural density of $\{n:P(n)<n^{\alpha},\,P(n+1)<(n+1)^{\beta}\}$ exists for all $\alpha,\beta\in(0,1)$ — and, for the independence question, that it equals $\rho(1/\alpha)\rho(1/\beta)$ — unconditionally, i.e. not assuming Elliott–Halberstam or any unproved hypothesis; or a proof that the natural density fails to exist for some $\alpha,\beta$. ADVANCES (checkable): prove the natural density exists (or equals the product) under a hypothesis strictly weaker than the Elliott–Halberstam input used by Wang [Wa21], clearly flagged; upgrade Teräväinen's logarithmic-density result [Te18] to natural density for a nontrivial range of $\alpha,\beta$; prove a case of the $r$-fold conjecture $\rho(1/\alpha)^r$ for some $r\geq 2$; or a certified high-range computation of the ratio $\#\{n\leq x:P(n)<n^\alpha,P(n+1)<(n+1)^\beta\}/x$ against $\rho(1/\alpha)\rho(1/\beta)$ with a reproducible program. Deliver the proof/formalisation or the reproducible computation with certified output.
Background
Posed by Erdős [Er76d], [Er76e, p.273], [ErPo78]; listed as open on erdosproblems.com/928 (fetched 2026-07-21, status 'open'), OEIS A006530 (largest prime factor). Background results: Dickman [Di30] showed the density of $n$ with $P(n)<n^{\alpha}$ is $\rho(1/\alpha)$, with $\rho$ the Dickman function. Meza observed that infinitely many $n$ meeting both conditions exist, via Schinzel's result [Sc67b] that for infinitely many $n$ the largest prime factor of $n(n+1)$ is at most $n^{O(1/\log\log n)}$. The recent frontier is substantial and must be respected: Teräväinen [Te18] proved that the logarithmic density of the set exists and equals $\rho(1/\alpha)\rho(1/\beta)$; and Wang [Wa21] proved the natural density equals $\rho(1/\alpha)\rho(1/\beta)$ conditional on the Elliott–Halberstam conjecture for friable (smooth) integers. What remains open is the existence of the natural (as opposed to logarithmic) density unconditionally. Erdős [Er76e] also conjectured the $r$-fold generalisation: the density of $n$ with $P(n+i)\leq n^{\alpha}$ for all $0\leq i<r$ is $\rho(1/\alpha)^r$. See also the companion Erdős #370 (erdosproblems.com/370). An attacker would bring the analytic machinery of friable integers in arithmetic progressions and short intervals (as in Teräväinen's and Wang's work) toward an unconditional natural-density statement, supported by high-range numerical computation of the counting function against $\rho(1/\alpha)\rho(1/\beta)\,x$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #928 (T. F. Bloom) | website |
| REF-02 | OEIS A006530 — largest prime factor of n | website |
| REF-03 | Erdős Problem #370 (companion; T. F. Bloom) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.