Erdős Problem #684

Precise statement

For the least $k$ at which the small-prime part of $\binom{n}{k}$ exceeds $n^2$, how large can $f(n)$ be?

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

Model not publicly disclosed

The source does not provide a sufficiently specific role description.

Provider: unknown · Prompt public: unknown · Independence: unknown

Verification boundary

contested

Key lemma refuted by a checked counterexample; see the claim issue.

Correctness: challenged · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-07-25

   erdosproblems.com/684

   VibeMathed reports this item as retracted. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

argument (source-reported)

Source-reported tools: argument.

Independent: unknown · difference confidence: 0

What remains uncertain

A preprint claimed $\limsup f(n)/\log n = \infty$, but a deterministic audit later found a counterexample to its key Lemma 18. The stated conclusion is not established and the problem remains open.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• A preprint claimed $\limsup f(n)/\log n = \infty$, but a deterministic audit later found a counterexample to its key Lemma 18. The stated conclusion is not established and the problem remains open.