The board / number theory / #685
Erdős Problem #685
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`primeRecipSum k n` is the sum of \(1/p\) over the primes \(k < p < n\).
Let \(\epsilon > 0\) and let \(n\) be large depending on \(\epsilon\). Is it true that for all \(n^{\epsilon} < k \leq n^{1-\epsilon}\) the number of distinct prime divisors of \(\binom{n}{k}\) is \[(1+o(1))\, k \sum_{k < p < n} \frac{1}{p}?\] The \(o(1)\) is uniform in \(k\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/685.Prizenone
Topicsnumber theory, primes, binomial coefficients
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/685
Attempts
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The live page status is OPEN, but it lists 1 partial claimed proof (Jake Weatherhead, submitted 2026-07-14) and “Currently working on this problem: None”; stopped under the mandatory claimed-proof col
SKIPPED