SWhat 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.
Scientific 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
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.
Contributors
- The named authors in the research record · SCIENCE: Source-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.
- F. Hausdorff · SCIENCE: Suggestion helping replace the original 25-fold integral with the five-dimensional identity in the revised Annalen account.
- J. Kürschák · SCIENCE: Suggestion acknowledged with Hausdorff in the original footnote on page 282.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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