{"schema":"vela.problem-packet.v0.1","problem":700,"statement":"Let\\[f(n)=\\min_{1<k\\leq n/2}\\textrm{gcd}\\left(n,\\binom{n}{k}\\right).\\] Characterise those composite $n$ such that $f(n)=n/P(n)$, where $P(n)$ is the largest prime dividing $n$. Are there infinitely many composite $n$ such that $f(n)>n^{1/2}$? Is it true that, for every composite $n$,\\[f(n) \\ll_A \\frac{n}{(\\log n)^A}\\]for every $A>0$?","status":"open","seam":"sealed","closureRoutes":[{"type":"witness","verifierKind":"min_binom_gcd","note":"extended minimal-gcd case table re-checked by the frozen verifier"},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_2db3437d1512c222","banked":"f(n)=min gcd(n,C(n,k)) recomputed for the cited cases (semiprime and prime-power families)","open":"characterize f(n) in general beyond the verified families.","dependents":1,"lease":null}],"attestations":[],"attempts":[{"id":"att_0f6ad7b281820ab3","kind":"partial_proof","claim":"#700 RE-ASSESSED after reading erdosproblems comments (Tao, StijnC): substantially LESS attractive. Part 1 declared 'hopeless/undoable' by StijnC (confirms our probe). Part 3 'likely very hard' (Tao). Part 2 ALREADY has TWO conditional infinite families (StijnC + Kadi Siigur) -- the only open piece is an UNCONDITIONAL family, which StijnC found nontrivial AND says 'would not add much insight'. Recommend SKIP.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_0d0843b89407d581","kind":"partial_proof","claim":"Erdős #700 (f(n)=min_{1<k<=n/2} gcd(n,C(n,k))): verified f(n) table to n=2000 + reproduced known facts + classification seeds, Opus-spot-confirmed. Codex built the table (n=4..2000) and checked: prime-power f(p^a)=p (30 rows, 0 fail), semiprime f(n)=n/P(n) (577 rows, 0 fail), composite n with f(n)>sqrt(n) (89 rows, 65 with equality, 24 residual non-equality). Opus independently recomputed f(n): f(30)=6 (official example), all semiprime/prime-power facts reproduce, and 5/5 sampled sqrt-residual rows (n=140,168,280,420,495) match f-value AND minimizers exactly. These REPRODUCE known structure and provide classification DATA; they do NOT settle any official #700 sub-question (those remain open).","grade":"partial_proof","gateStatus":"verified","superseded":false}],"velaLean":[],"oeis":[{"id":"A091963","name":"a(n) is the smallest gcd of two interior numbers on row n of Pascal's triangle (\"interior\" means that the 1's at the ends of the rows are excluded).","terms":"2,3,2,5,2,7,2,3,2,11,3,13,2,3,2,17,2,19,4,3,2,23,3,5,2,3,4,29,6,31,2,3,2,5,4,37,2,3,5,41,6,43,4,3,2,47,3,7,2,3,4,53,2,5,","url":"https://oeis.org/A091963"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}