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.

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).

Order-nr.: 41656