{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1245",
  "content_hash": null,
  "slug": "erdos-problem-51-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\\varphi$ denote Euler's totient function, which counts the number of integers in $\\{1, 2, \\ldots, n\\}$ that are coprime to $n$. For a positive integer $a$, let $\\varphi^{-1}(\\{a\\})$ denote the set of all positive integers $n$ such that $\\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\\varphi^{-1}(\\{a\\})$ is nonempty for every $a \\in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \\subseteq \\mathbb{N}$ and a function $n : A \\to \\mathbb{N}$ such that for every $a \\in A$, the integer $n(a)$ is the least element of $\\varphi^{-1}(\\{a\\})$, and $\\displaystyle\\lim_{\\substack{a \\to \\infty \\\\ a \\in A}} \\frac{n(a)}{a} = \\infty$?",
  "relevance": "For Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages, record erdos-problem-51-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\\varphi$ denote Euler's totient function, which counts the number of integers in $\\{1, 2, \\ldots, n\\}$ that are coprime to $n$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\\varphi$ denote Euler's totient function, which counts the number of integers in $\\{1, 2, \\ldots, n\\}$ that are coprime to $n$. For a positive integer $a$, let $\\varphi^{-1}(\\{a\\})$ denote the set of all positive integers $n$ such that $\\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\\varphi^{-1}(\\{a\\})$ is nonempty for every $a \\in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \\subseteq \\mathbb{N}$ and a function $n : A \\to \\mathbb{N}$ such that for every $a \\in A$, the integer $n(a)$ is the least element of $\\varphi^{-1}(\\{a\\})$, and $\\displaystyle\\lim_{\\substack{a \\to \\infty \\\\ a \\in A}} \\frac{n(a)}{a} = \\infty$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/51",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/51",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1246",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-51",
      "title": "erdos problem 51",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}