{"schema":"vela.problem-packet.v0.1","problem":458,"statement":"Let $[1,\\ldots,n]$ denote the least common multiple of $\\{1,\\ldots,n\\}$. Is it true that, for all $k\\geq 1$,\\[[1,\\ldots,p_{k+1}-1]&#60; p_k[1,\\ldots,p_k]?\\]","status":"open","seam":"sealed","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_ecab16a7fa140c31","kind":"reduction","claim":"lcm(1..p_{k+1}-1) < p_k*lcm(1..p_k)  <=>  P(k) := prod of bases q of prime powers q^m (m>=2) in the gap (p_k,p_{k+1}) is < p_k.","grade":"verified_reduction","gateStatus":"needs_verification","superseded":false},{"id":"att_e2a9820ea9ec786a","kind":"dead_end","claim":"#458 superseded finite-range proof attempt: the reduction to prime-power bases is useful, but the claimed BHP-driven unconditional range was overstrong and is replaced by att_1f15d7e468458e15.","grade":"honest_null","gateStatus":"superseded","superseded":false},{"id":"att_cbc707ca85bc389b","kind":"dead_end","claim":"#458 superseded cross-model finite-range claim: the shared reduction and bounded searches were useful, but the claimed unconditional p<1.21e24 conclusion is not accepted frontier state.","grade":"honest_null","gateStatus":"superseded","superseded":false},{"id":"att_829a69463ae826c3","kind":"partial_proof","claim":"Erdos #458: exact lcm reduction to P(p,r)=prod of bases of prime powers in the gap (p,r) < p; bounded-verified to 1e12, no counterexample. OPEN: two prime squares in one gap is a SUFFICIENT obstruction, but the converse is unproved; mixed high-power clusters remain a real proof gap.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_1f15d7e468458e15","kind":"research_audit","claim":"Erdos #458 remains open: Vela now has a named Lean target, checked gap-support lemmas, exact prime-power gap verification through 10^15 with no counterexample, finite square/two-cube obstruction bridges, and narrowed mixed-cluster residues.","grade":"obstruction_map","gateStatus":"verified","superseded":false},{"id":"att_6fba8db9c5992ca7","kind":"dead_end","claim":"#458 superseded obstruction-classification attempt: the sigma=sum 1/a_i filter is useful, but this record overclaimed complete classification and is replaced by att_1f15d7e468458e15.","grade":"obstruction_map","gateStatus":"superseded","superseded":false}],"velaLean":[{"file":"lean/Vela/Erdos458.lean","sorryFree":true,"url":"https://github.com/vela-science/vela-internal/blob/main/lean/Vela/Erdos458.lean"}],"oeis":[{"id":"A056604","name":"a(0)=1; thereafter a(n) = lcm(1, 2, 3, 4, ..., prime(n)).","terms":"1,2,6,60,420,27720,360360,12252240,232792560,5354228880,2329089562800,72201776446800,5342931457063200,219060189739591200","url":"https://oeis.org/A056604"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}