CHAP. XXV] WAKING'S PROBLEM. 723 



f(x+h) >0 for x real [cf. Hurwitz 44 ], since 



being true for f(x+h)=h". Hilbert's lemma is proved by use of 



whence follows Hurwitz's theorem that Waring's theorem is true for n = 2m 

 if true for n = m. Finally, he simplified the second (elementary) part of 

 Hilbert's proof of Waring's theorem. 

 A. Boutin 37 gave the identities 



^(xyzuY = 192xyzu, db(3i- -x n ) n = nl2 n x 1 - - >x n , 



8 2" 



the exterior sign being the product of the n interior signs. 



P. Bachmann 38 gave an exposition of several of the preceding papers. 



A. Fleck 38a and W. Wolff 386 proved that every definite quartic function 

 of x with rational coefficients is a sum of five squares of rational integral 

 functions with rational coefficients. 



E. Landau 39 gave a new elementary proof that all numbers exceeding a 

 certain limit and prime to 10 (or to the product of any two primes of the 

 form 3m + 2) are sums of at most 8 positive cubes. He here avoided the 

 theory of the distribution of primes used in his 32 former proof. 



J. Kiirscha'k 40 generalized Liouville's 6 identity (1) to give 



(3/c\ 

 k } (oJ 



where on the left occur all possible combinations of signs and all sets of 

 of the 3&+1 variables o , , a sk > For ra^3, there is no identity 



A. Ge'rardin 41 noted that (a: 3 +9i/ 3 ) 3 is the sum of the cubes of x 3 , y 3 , 

 , 8?/ 3 , 3x 2 y, 3xy*, Qxy 2 . Also (x 3 +3y 3 ) 3 is the sum of the cubes of x 3 , 

 3y 3 , 3xy 2 , 2x 2 y, x 2 y. L. Rouve remarked that the former is the sum of the 

 cubes of x 3 , 3x 2 y, $y 3 , 3xy 2 , Qxy 2 . 



A. J. Kempner 42 considered the number C(k, n) of the positive integers 

 ^k which are sums of n or fewer positive nth powers, and the superior 

 limit S of C(k, ri)/k for k= . He proved that S<l{n\, whereas Hurwitz 28 

 and Maillet 27 had proved merely that S<1. It follows that there is an 

 infinitude of positive integers of each of the forms 9Z, 9Z+1, , 9Z+8, such 

 that each is not a sum of fewer than four positive cubes. There is an 

 infinitude of positive integers each not a sum of fewer than nine sixth 



37 L'intermSdiaire des math., 17, 1910, 122-3, 236-7. See papers 66-68 below. 



38 Niedere Zahlentheorie, 2, 1910, 328-48. 



Archiv Math. Phys., (3), 10, 1906, 23-38; (3), 16, 1910, 275-6. 

 386 Vierteljahreschrift Naturf. Gesell. Zurich, 56, 1911, 110-24. 



39 Archiv Math. Phys., (3), 18, 1911, 248-252. 

 Ibid., 242-3. 



41 Sphinx-Oedipe, 6, 1911, 19, 95. 



ber das Waringsche Problem und einige Verallgemeinerungen, Disa., Gottingen, 1912. 

 Extract in Math. Annalen, 72, 1912, 387. 



