{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R808",
  "content_hash": null,
  "slug": "tspu-claim-almkvist-r3",
  "type": "claim",
  "title": "Almkvist's theorem settles every n at least 11",
  "summary": "The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.",
  "relevance": "For Eventual unimodality of ternary subset-sum polynomials, record tspu-claim-almkvist-r3 (“Almkvist's theorem settles every n at least 11”) records a bound, answer, status fact, or structural consequence. The record states: The candidate is the r = 3 case of a unimodality theorem proved by Gert Almkvist in 1989.",
  "relevance_source": "recorded",
  "body": "For every positive integer \\(n\\),\n\\[\nP_n(q)=\\prod_{k=1}^n(1+q^k+q^{2k})\n      =\\prod_{k=1}^n\\frac{1-q^{3k}}{1-q^k}.\n\\]\nAlmkvist studied the more general polynomial\n\\[\nf_{n,r}(q)=\\prod_{k=1}^n\\frac{1-q^{rk}}{1-q^k}.\n\\]\nHis 1989 paper proves the conjectured unimodality for \\(3\\leq r\\leq20\\), as well as \\(r=100,101\\), with the odd-\\(r\\) range beginning at \\(n=11\\). Setting \\(r=3\\) gives the candidate verbatim. Hence the coefficient sequence of \\(P_n\\) is unimodal for every \\(n\\geq11\\).\n\nDong and Ji identify the result explicitly in their introduction: their Conjecture 1.1 is the displayed product, and the paragraph after equation (1.5) records Almkvist's proved cases \\(3\\leq r\\leq20\\). The candidate is a rediscovery of this classical result.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every integer n at least 11"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0022-314X(89)90096-6",
      "locator": "G. Almkvist, Proof of a conjecture about unimodal polynomials, Journal of Number Theory 32 (1989), 43-57, r = 3 specialization; cross-identified in Dong and Ji, Unimodality of partition polynomials related to Borwein's conjecture, Introduction, Conjecture 1.1 and the paragraph after equation (1.5)"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0022-314X(89)90096-6",
    "locator": "G. Almkvist, Proof of a conjecture about unimodal polynomials, Journal of Number Theory 32 (1989), 43-57, r = 3 specialization; cross-identified in Dong and Ji, Unimodality of partition polynomials related to Borwein's conjecture, Introduction, Conjecture 1.1 and the paragraph after equation (1.5)"
  },
  "relations": [
    {
      "slug": "R809",
      "title": "The complete failure set is 2, 4, 6, 8, 9, and 10",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R807",
      "title": "Exact coefficient sweep through n = 300",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "ternary-subset-polynomial-unimodality",
      "title": "ternary subset polynomial unimodality",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}