cb-0043.json

{
  "format_version": "1.0",
  "id": "cb-0043",
  "slug": "lehmer-totient",
  "title": "Lehmer totient",
  "field": "number-theory-and-discrete-mathematics",
  "statement": "If φ(n) divides n−1, then n is prime.",
  "counterexample": {
    "description": "A composite \\(n\\) satisfying \\(\\varphi(n)\\mid n-1\\). A complete factorization makes compositeness, \\(\\varphi(n)\\), and divisibility cheap to verify.",
    "finite": true
  },
  "status_observation": {
    "state": "open",
    "as_of": "2026-07-22",
    "evidence_state": "source-linked"
  },
  "provenance": {
    "collection": "original-100-inventory",
    "sources": [
      {
        "url": "https://ems.press/journals/em/articles/8549048",
        "role": "supporting"
      }
    ]
  }
}