Erdős (Erdos) Problem #983: status in 2026

Erdős Problem #983

Let n≥ 2 and π(n)<k≤ n. Let f(k,n) be the smallest integer r such that in any A⊆ 1,…,n of size lvert Arvert=k there exist primes p_1,…,p_r such that >r many a∈ A are only divisible by primes from p_1,…,p_r. In general, estimate f(k,n), particularly when π(n)+1<k=o(n). This is the second part.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-983,
  title        = {Erdős Problem #983},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-983}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Let $n \geq 2$ and $π (n) < k \leq n$ . Let $f (k, n)$ be the smallest integer $r$ such that in any $A \subseteq {1, \dots, n}$ of size $∣ A ∣ = k$ there exist primes $p_{1}, \dots, p_{r}$ such that $> r$ many $a \in A$ are only divisible by primes from ${p_{1}, \dots, p_{r}}$ .

In general, estimate $f (k, n)$ , particularly when $π (n) + 1 < k = o (n)$ .

This is the second part. We read "estimate" as an asymptotic formula answer k n, valid for every function $k = k (n)$ with $π (n) + 1 < k (n) = o (n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«983».

theorem erdos_983.parts.ii :
    ∀ k : ℕ → ℕ, (∀ᶠ n : ℕ in atTop, Nat.primeCounting n + 1 < k n) →
      ((fun n : ℕ ↦ (k n : ℝ)) =o[atTop] (fun n : ℕ ↦ (n : ℝ))) →
      (fun n : ℕ ↦ (f (k n) n : ℝ)) ~[atTop]
        (fun n : ℕ ↦ erdos_983.parts.ii.answer (k n) n)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/983. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/983
[Er70b] Erdős, P., Some applications of graph theory to number theory. Proc. Second Chapel Hill Conf. on Combinatorial Mathematics and its Applications (Univ. North Carolina, Chapel Hill, N.C., 1970) (1970), 136-145. PDF
[Po79] Pomerance, C., The prime number graph. Math. Comp. 33 (1979), 399-408. PDF

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.