Erdős problems frontier

{
 "assertion": {
  "direction": null,
  "entities": [],
  "relation": null,
  "text": "Erdős Problem #442 has been DISPROVED (a counterexample is known). Statement: Let $\\operatorname{Log} x := \\max\\{\\log x, 1\\}$, $\\operatorname{Log}_2x = \\operatorname{Log} (\\operatorname{Log} x)$, and $\\operatorname{Log}_3x = \\operatorname{Log}(\\operatorname{Log}(\\operatorname{Log} x)).$ Is it true that if $A\\subseteq\\mathbb{N}$ is such that $$ \\frac{1}{\\operatorname{Log}_2 x} \\sum_{n\\in A: n\\leq x} \\frac{1}{n}\\to\\infty $$ then $$ \\left(\\sum_{n\\in A: n\\leq x} \\frac{1}{n}\\right)^2 \\sum_{n, m \\in A: n < m \\leq x} \\frac{1}{\\operatorname{lcm}(n, m)}\\to\\infty $$ as $x\\to\\infty$? Tao [Ta24b] has shown this is false. [Ta24b] Tao, T., _Dense sets of natural numbers with unusually large least common multiples_. arXiv:2407.04226 (2024). Note: the informal and formal statements follow the solution paper https://arxiv.org/pdf/2407.04226 Topics: number theory. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.",
  "type": "theoretical"
 },
 "conditions": {
  "age_group": null,
  "cell_type": null,
  "clinical_trial": false,
  "concentration_range": null,
  "duration": null,
  "human_data": false,
  "in_vitro": false,
  "in_vivo": false,
  "species_unverified": [],
  "species_verified": [],
  "text": "Agent-imported candidate claim; scope requires review."
 },
 "confidence": {
  "basis": "agent-imported candidate claim; reviewer acceptance required",
  "extraction_confidence": 0.7,
  "kind": "frontier_epistemic",
  "method": "expert_judgment",
  "score": 0.99
 },
 "created": "2026-05-30T00:42:06.826507+00:00",
 "evidence": {
  "effect_size": null,
  "evidence_spans": [
   {
    "artifact_id": "va_9bc926d75e4e3881",
    "artifact_packet_id": "cap_61973ee16b553d57",
    "candidate_claim_id": "vc_e18a1401889f4ca1"
   }
  ],
  "method": "ScienceClaw-shaped artifact packet import",
  "model_system": "agent artifact packet",
  "p_value": null,
  "replicated": false,
  "replication_count": null,
  "sample_size": null,
  "species": null,
  "type": "computational"
 },
 "flags": {
  "contested": false,
  "declining": false,
  "gap": false,
  "gravity_well": false,
  "negative_space": false,
  "retracted": false
 },
 "id": "vf_b3764125348b0dbb",
 "links": [],
 "previous_version": null,
 "provenance": {
  "authors": [
   {
    "name": "Erdős Open-Problem spine ingest",
    "orcid": null
   }
  ],
  "citation_count": null,
  "doi": null,
  "extraction": {
   "extracted_at": "2026-05-30T00:42:06.826507+00:00",
   "extractor_version": "vela/0.691.0",
   "method": "artifact_to_state_import",
   "model": "agent:erdos-spine-ingest",
   "model_version": null
  },
  "journal": null,
  "openalex_id": null,
  "pmc": null,
  "pmid": null,
  "publisher": "artifact packet",
  "review": {
   "corrections": [],
   "reviewed": false,
   "reviewed_at": null,
   "reviewer": null
  },
  "source_type": "model_output",
  "title": "cap_61973ee16b553d57 · vc_e18a1401889f4ca1",
  "url": "https://www.erdosproblems.com/442",
  "year": null
 },
 "updated": null,
 "version": 1
}