Waring's Problem Solved (1 results)
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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. First printing of a groundbreaking work in Number Theory. Edward Waring (1734-98) stated, i…n 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).