What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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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People and systems
David Hilbert
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
Suggestion helping replace the original 25-fold integral with the five-dimensional identity in the revised Annalen account.
J. Kürschák
Suggestion acknowledged with Hausdorff in the original footnote on page 282.
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English catalogue narrative
Hilbert: a finite number of kth powers suffices for every positive integer
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
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.
Reading status
This 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.
People and systems
David Hilbert
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.