Two integers between consecutive primes with all prime factors below the gap, infinitely often (Erdős #932)
Statement
Let $p_k$ denote the $k$th prime. Prove or disprove: for infinitely many $r$ there are at least two integers $n$ with $p_r < n < p_{r+1}$ all of whose prime factors are less than the prime gap $p_{r+1}-p_r$. (Equivalently: infinitely often the open interval between consecutive primes contains two distinct $(p_{r+1}-p_r)$-smooth integers.)
Acceptance. FULLY RESOLVES: a complete proof — machine-checkable preferred, else a full written proof — that infinitely many $r$ admit at least two integers in $(p_r,p_{r+1})$ that are $(p_{r+1}-p_r)$-smooth; OR a disproof, i.e. a proof that only finitely many such $r$ exist. ADVANCES: prove the weaker statement that infinitely many $r$ admit at least one such $n$; extend the exhaustive computation (e.g. the OEIS A387864 census) to a new record range with reproducible code and a certificate of completeness over that range; or improve the counting/density bound on the exceptional set strictly beyond Erdős's density-$0$ result, with proof. Deliver the proof, or the search code together with the attained range and the tabulated good gaps.
Background
Posed by Erdős [Er76d]. Erdős believed the statement is true but that such $r$ are rare: he proved that the set of $r$ for which even a single such $n$ exists has natural density $0$. Thus a positive-density phenomenon is ruled out, and the open content is that the count of good $r$ is nonetheless infinite — indeed with (at least) two smooth integers packed into the same gap. On the site Terence Tao flagged the problem as difficult. A related census is catalogued as OEIS A387864. Formalised in Lean as part of the Google DeepMind Formal Conjectures project (formal-conjectures/932). Listed as open on erdosproblems.com/932 (fetched 2026-07-21, status 'open'); no Erdős prize is attached. Attacker's tool: large-scale computation over prime gaps (sieve each interval $(p_r,p_{r+1})$, factor its members, test smoothness against the gap length) to extend the associated OEIS census and gather statistical support, combined with analytic sieve and prime-gap methods (Maier-type matrices, smooth-numbers-in-short-intervals results) toward a proof.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #932 (T. F. Bloom) | website |
| REF-02 | OEIS A387864 | website |
| REF-03 | Lean formalisation of Erdős #932 (Google DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.