{"schema":"vela.problem-packet.v0.1","problem":686,"statement":"Can every integer $N\\geq 2$ be written as\\[N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}\\]for some $k\\geq 2$ and $m\\geq n+k$?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_d76a51a8a4ff297a","kind":"partial_proof","claim":"Erdos #686: lean/Vela/Erdos686.lean (kernel-checked, axiom-clean, no sorry) proves the k=2 and k=4 obstructions for prime-power squares. If A is a prime power then A^2 has no representation N=prod_{i<=k}(m+i)/prod_{i<=k}(n+i) (m>=n+k) with k=2 or k=4 -- in particular N=4 has none -- via prime-power divisibility (no multiple of A lies strictly between consecutive multiples A*N0 and A*(N0+1)) plus a general k=4 => k=2 reduction with the same multiplier ((t+1)..(t+4)=4*(u+1)(u+2)). This is theorem-grade partial progress; it does NOT address k=3, k>=5, or composite non-prime-power square N.","grade":"partial_proof","gateStatus":"verified","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos686Tail.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos686Tail.lean"},{"file":"lean/Vela/Erdos686.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos686.lean"}],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}