Erdős Problem #699 — open problem in number theory

Erdős Problem #699

Erdős Problem 699. Is it true that for every 1 ≤ i < j ≤ n / 2 there exists a prime p ≥ i with p | gcd(C(n, i), C(n, j))?

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.

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Cite

@misc{cairn-erdos-699,
  title        = {Erdős Problem #699},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-699}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_699. Erdős Problem 699. Is it true that for every $1 \leq i < j \leq n /2$ there exists a prime $p \geq i$ with $p ∣ g cd ((i n), (j n))$ ?

erdos_szekeres_strengthening. Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j), one can always take p > i in the prime divisor statement.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«699» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_699 : answer(sorry) ↔
    ∀ n i j : ℕ,
      1 ≤ i →
      i < j →
      j ≤ n / 2 →
      ∃ p : ℕ, p.Prime ∧ i ≤ p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j)

theorem erdos_szekeres_strengthening : answer(sorry) ↔
    ∃ E : Finset (ℕ × ℕ × ℕ), ∀ n i j : ℕ,
      1 ≤ i →
      i < j →
      j ≤ n / 2 →
      (n, i, j) ∉ E →
      ∃ p : ℕ, p.Prime ∧ i < p ∧ p ∣ Nat.gcd (Nat.choose n i) (Nat.choose n j)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/699. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/699

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.