‎"HILBERT, DAVID.‎
‎Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem). Dem Andenken an Hermann Minkowski Gewidmet. - [Hilbert-Kamke problem]‎

‎Leipzig, B. G. Teubner, 1909. 8vo. Bound in contemporary half calf with gilt lettering to spine. In ""Mathematische Annalen"", 67 band. 1909. Bookplates to pasted down front free end-paper and library stamp to verso of title page. Top half of spine is detached. Bookblock, however, still firmly attached. Fine and clean. Pp. 281-300. [Entire volume: IV, 575 pp.].‎

Reference : 47248

‎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).‎