‎"HILBERT, DAVID. - WARING'S PROBLEM SOLVED.‎
‎Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl nter Potenzen (Waringische Problem).‎

‎Leipzig, B.G. Teubner, 1909. Orig. printed wrappers. No backstrip. In. ""Mathematische Annalen. Hrsg. von Felix Klein, Walther v. Dyck, David Hilbert, Otto Rosenthal"", 67. Bd., 3. Heft. Pp. 281-432 (=3. Heft). Hilbert's paper: pp. 281-300.‎

Reference : 41656

‎First printing of a groundbreaking work in Number Theory. Edward Waring (1734-98) stated, in his ""Meditationes Algebraicae"" (1770), the theorem known now as ""Waring's Theorem"", that every integer is either a cube or the sum of at most nine cubes"" also every integer is either a fourth power of the sum of at most 19 fourth powers. He conjectured also that every positive integer can be expressed as the sum of at most r kth powers, the r depending on k. These theoremes were not proven by him, but by David Hilbert in the paper offered.Hilbert proves that for every integer n, there exists an integer m such that every integer is the sum of m nth powers. This expands upon the hypotheis of Edward Waring that each positive integer is a sum of 9 cubes (n=3, m=9) and of 19 fourth powers (n= 4, m=19).This issue also contains F. Hausdorff's ""Zur Hilbertschen Lösung des Waringschen Problems"", pp. 301-305.(Se Kline p. 609).‎