{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1193",
  "content_hash": null,
  "slug": "erdos-problem-124-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 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that\n\\[\\sum_{d \\in D} \\frac{1}{d-1} \\geq 1\\]\nand $\\gcd(D) = 1$. For each $d \\in D$ and each integer $k \\geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as\n\\[\\sum_{i \\in s} d^i\\]\nfor some finite set $s$ of integers with $i \\geq k$ for all $i \\in s$. Does every sufficiently large natural number belong to the set\n\\[\\sum_{d \\in D} S(d,k),\\]\nthat is, can it be expressed as $\\sum_{d \\in D} x_d$ where $x_d \\in S(d,k)$ for each $d \\in D$?",
  "relevance": "For Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift, record erdos-problem-124-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 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that\n\\[\\sum_{d \\in D} \\frac{1}{d-1} \\geq 1\\]\nand $\\gcd(D) = 1$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that\n\\[\\sum_{d \\in D} \\frac{1}{d-1} \\geq 1\\]\nand $\\gcd(D) = 1$. For each $d \\in D$ and each integer $k \\geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as\n\\[\\sum_{i \\in s} d^i\\]\nfor some finite set $s$ of integers with $i \\geq k$ for all $i \\in s$. Does every sufficiently large natural number belong to the set\n\\[\\sum_{d \\in D} S(d,k),\\]\nthat is, can it be expressed as $\\sum_{d \\in D} x_d$ where $x_d \\in S(d,k)$ for each $d \\in D$? 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/124",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/124",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1194",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-124",
      "title": "erdos problem 124",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}