{"schema":"vela.problem-packet.v0.1","problem":931,"statement":"Let $k_1\\geq k_2\\geq 3$. Are there only finitely many $n_2\\geq n_1+k_1$ such that\\[\\prod_{1\\leq i\\leq k_1}(n_1+i)\\textrm{ and }\\prod_{1\\leq j\\leq k_2}(n_2+j)\\]have the same prime factors?","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_0167f047170ab324","kind":"partial_proof","claim":"erdos_931 (finiteness of same-radical consecutive-product pairs) is genuinely open and remains so. Real partial work: showed abc gives rad of 3 consecutive ints ≫ m^{2−ε} but does NOT yield finiteness","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_f66f51688b8f230b","kind":"reduction","claim":"Erdős #931 reduces (via Størmer) to: for fixed (k1,k2), are there only finitely many admissible GAPS d=n2-n1? Plus an INDEPENDENTLY VERIFIED 26-pair certificate for official (4,3), and a correction: the published erdosproblems forum table for (4,3) is INVALID (composite 'prime' entries, non-verifying rows) though its count of 26 is correct.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_cf59fd4dcb6bb38c","kind":"partial_proof","claim":"Deep (4,3) structural framework (cross-model verified): exact prime-local condition + incidence-matrix + a NEW pure congruence obstruction d != 4 (mod 7), verified on all 26 known pairs. Reduces the (4,3) case to a concrete wedge lemma (finiteness of empty-row/column solutions).","grade":"obstruction_map","gateStatus":"needs_verification","superseded":false},{"id":"att_28afa6de7420e6f9","kind":"partial_proof","claim":"Lemma-D push (verified): proved a SPARSITY bound — empty-row/column (4,3) solutions number << exp((3log2+o(1)) logX/loglogX) = X^{o(1)} — but NOT finiteness (the target). Built on the standard Stormer-Pell mechanism and a JULY-2025 paper directly on #931 (Lebowitz-Lockard arXiv:2507.09899), which records (k,l)>1 finiteness as an OPEN CONJECTURE. So #931 is at the active research frontier, not under-defended.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false},{"id":"att_e9ea6758fbad444a","kind":"partial_proof","claim":"Stronger conditional finiteness (verified): bounded-cofactor theorem — for any fixed K, only finitely many (4,3) solutions have an active B_j with a prime factor p>=5 of bounded cofactor (B_j=qp, q<=K). So any INFINITE (4,3) family must be deeply friable: P+(B_j)/B_j -> 0 for all active columns. NOT finiteness.","grade":"partial_proof","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}