Estimate $f(k)$: the longest run of $k$-smooth consecutive integers above $k$ (Erdős #961)
Statement
Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers greater than $k$ contains an integer divisible by some prime $p>k$. Equivalently, $f(k)-1$ is the maximal length of a block of consecutive integers, each greater than $k$ and each $k$-smooth (having all prime factors $\leq k$). Estimate $f(k)$.
Acceptance. FULLY RESOLVES: a complete proof pinning down the order of growth of $f(k)$ — for instance a proof that $f(k)\ll (\log k)^{O(1)}$, or a matching lower bound, or a refutation — machine-checkable (Lean) preferred, else a full rigorous written proof. ADVANCES (each independently checkable): a proof of an asymptotic upper bound strictly beating the best stated bound $f(k)\ll \frac{\log\log\log k}{\log\log k}\frac{k}{\log k}$ (Jutila; Ramachandra–Shorey); a proof of a new nontrivial lower bound on $f(k)$; or an extended exact table of $f(k)$ over a stated range of $k$, produced with reproducible code and a certificate that each tabulated value is exhaustively correct. Deliver the proof, or the computation code plus the certified table.
Background
Posed by Erdős [Er76e, p.271]; listed as open on erdosproblems.com/961 (fetched 2026-07-21, status 'open'). The question asks how long a run of consecutive $k$-smooth integers can be. Sylvester and Schur (see [Er34]) proved $f(k)\leq k$. Erdős [Er55d] proved $f(k)<3k/\log k$. Jutila [Ju74] and, independently, Ramachandra and Shorey [RaSh73] proved $$f(k)\ll \frac{\log\log\log k}{\log\log k}\cdot\frac{k}{\log k},$$ the best known upper bound. It is believed that in fact $f(k)\ll (\log k)^{O(1)}$. The problem is essentially equivalent to Erdős #683 (erdosproblems.com/683), which SciNet already hosts as a question on the largest prime factor of $\binom{n}{k}$; the present formulation studies the object $f(k)$ directly. The statement has a Lean formalization (formal-conjectures), and OEIS A213253 records related data. Attacker's tool: exact computation of $f(k)$ for small $k$ by enumerating maximal runs of $k$-smooth integers, tabulating the results against the conjectured $(\log k)^{O(1)}$ growth, together with sieve-analytic refinements of the Jutila / Ramachandra–Shorey bound.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #961 (T. F. Bloom) | website |
| REF-02 | OEIS A213253 — related to runs of smooth consecutive integers | website |
| REF-03 | Lean formalization (formal-conjectures #961) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.