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    "title": "Hilbert: a finite number of kth powers suffices for every positive integer",
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        "title": "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)",
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        "description": "An existence theorem with the quantifier order for every k, some s(k), every positive integer; it does not identify the least possible bound g(k), give a common bound independent of k or establish optimal asymptotic Waring constants. Negative bases/signed cancellation are not a replacement for the positive-summand conclusion. The Mathematische Annalen version contains changes and additions relative to the 1909 Göttingen report and credits Hausdorff/Kürschák suggestions and earlier special-exponent proofs. The source says bounds can be obtained from the argument, so this record does not call the result inherently ineffective.",
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          "citation": "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)",
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      "statement": "# Hilbert: a finite number of kth powers suffices for every positive integer\n\nEvery positive integer is a sum of a bounded number of squares, but what happens for cubes, fourth powers or any other fixed exponent? Hilbert proves that each positive integer exponent has its own finite bound on the number of positive-power summands needed for every positive integer. His 1909 proof turns an integral identity into finite rational identities and then integer representations. It establishes existence for all exponents, without finding the least possible bound.\n\n## Scientific scope\n\nAn existence theorem with the quantifier order for every k, some s(k), every positive integer; it does not identify the least possible bound g(k), give a common bound independent of k or establish optimal asymptotic Waring constants. Negative bases/signed cancellation are not a replacement for the positive-summand conclusion. The Mathematische Annalen version contains changes and additions relative to the 1909 Göttingen report and credits Hausdorff/Kürschák suggestions and earlier special-exponent proofs. The source says bounds can be obtained from the argument, so this record does not call the result inherently ineffective.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nDavid Hilbert\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)](https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0067/LOG_0029.pdf)\n",
      "summary": "Every positive integer is a sum of a bounded number of squares, but what happens for cubes, fourth powers or any other fixed exponent? Hilbert proves that each positive integer exponent has its own finite bound on the number of positive-power summands needed for every positive integer. His 1909 proof turns an integral identity into finite rational identities and then integer representations. It establishes existence for all exponents, without finding the least possible bound.",
      "title": "Hilbert: a finite number of kth powers suffices for every positive integer"
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    "title": "Hilbert: a finite number of kth powers suffices for every positive integer",
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    "what_changed": "Every positive integer is a sum of a bounded number of squares, but what happens for cubes, fourth powers or any other fixed exponent? Hilbert proves that each positive integer exponent has its own finite bound on the number of positive-power summands needed for every positive integer. His 1909 proof turns an integral identity into finite rational identities and then integer representations. It establishes existence for all exponents, without finding the least possible bound.",
    "scope": "Every positive integer is a sum of a bounded number of squares, but what happens for cubes, fourth powers or any other fixed exponent? Hilbert proves that each positive integer exponent has its own finite bound on the number of positive-power summands needed for every positive integer. His 1909 proof turns an integral identity into finite rational identities and then integer representations. It establishes existence for all exponents, without finding the least possible bound. An existence theorem with the quantifier order for every k, some s(k), every positive integer; it does not identify the least possible bound g(k), give a common bound independent of k or establish optimal asymptotic Waring constants. Negative bases/signed cancellation are not a replacement for the positive-summand conclusion. The Mathematische Annalen version contains changes and additions relative to the 1909 Göttingen report and credits Hausdorff/Kürschák suggestions and earlier special-exponent proofs. The source says bounds can be obtained from the argument, so this record does not call the result inherently ineffective.",
    "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. An existence theorem with the quantifier order for every k, some s(k), every positive integer; it does not identify the least possible bound g(k), give a common bound independent of k or establish optimal asymptotic Waring constants. Negative bases/signed cancellation are not a replacement for the positive-summand conclusion. The Mathematische Annalen version contains changes and additions relative to the 1909 Göttingen report and credits Hausdorff/Kürschák suggestions and earlier special-exponent proofs. The source says bounds can be obtained from the argument, so this record does not call the result inherently ineffective.",
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        "title": "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem)",
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