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Erdős Problem #684

Erdős problem #684 · erdosproblems.com/684

For the least kk at which the small-prime part of (nk)\binom{n}{k} exceeds n2n^2, how large can f(n)f(n) be?

Result
Proved
Status
Retracted
AI contribution
Method
Argument
Field
Number Theory, Binomial Coefficients
Posed by
Year posed
Years open
Solved
2026-07-25
Model
Model not publicly disclosed
Vendor
Collaborators
Verification
Contested
Publication
Announced
Significance
10 / 100
Disclosed cost
Wikipedia
No dedicated article

Verification

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

Claim issue

A preprint claimed lim supf(n)/logn=\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.

Source

erdosproblems.com/684

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