724 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxv 



powers, and an infinitude each not a sum of fewer than 2 9+2 powers, with 

 the exponent 2 5 , for g>l. He lowered the known limit for N 6 to 970 by 

 use of the identity 



8 12 4 



for c = d and for d = 0, and the fact that every number is of one of the forms 

 a 2 + & 2 + kc 2 f or k = 1 , 2 . For the determination of upper limits for Nu and Nu 

 from known limits for N& and NT, he gave identities expressing Z(a 2 +6 2 +c 2 ) n 

 as a sum of (2n)th powers, for n = 6 and 7, where I is a suitably chosen 

 integer. 



R. Remak 43 noted that Stridsberg 36 used integrals in a single place and 

 applied the result proved by them only for the special case hi which 

 f(a)=g 2 (a). For this case Remak gave an elementary proof by use of the 

 fact that a quadratic form in n variables is definite if the determinant of the 

 part involving the first v variables (suitably chosen) is positive for v = l, 2, 

 , n. Hence the proof of Waring's theorem is reduced to algebraic pro- 

 cesses. 



A. Hurwitz 44 gave a new elementary proof of the theorem, used by 

 Hausdorff, 35 Stridsberg 36 and Remak, 43 that if the real polynomial 



f(x) = C +CiOH ----- \-C 2n X 2n , 



not identically zero, is ^0 for every real x } then 



is positive for every real x; likewise f or /(#) +/'(#) + 



L. Orlando 440 amplified Hurwitz's 44 proof. 



G. Frobenius 45 also gave an algebraic proof of Waring's theorem by 

 altering Stridsberg's proof at the point where he had used integrals. 



E. Schmidt 46 used Minkowski's convex point sets in space of q dimen- 

 sions to give a more luminous exposition of Hilbert's first lemma. 



G. Loria 47 remarked that if Waring's minimum 19 for AT 4 could be 

 lowered to 16 [^overlooking the facts noted by J. A. Euler], one would hope 

 for a proof that every number is a sum of n z exact nth powers. 



E. Landau 48 pointed out errors in the same journal on sums of cubes. 



W. S. Baer 49 proved that every integer =s23 10 14 is a sum of 8 or fewer 

 positive cubes, likewise every odd number >175 396 368 704, and every 

 number =8 (mod 16). The following numbers are sums of 7 or fewer posi- 

 tive cubes: every number 2744s (s odd), all sufficiently large multiples of 

 16 or 27, all sufficiently large numbers =0, 8, 16, 24, 28, 36, 44, 48, 56, 64 



" Math. Annalen, 72, 1912, 153-6. 



44 Ibid., 73, 1912, 173-8. Cf. Orlando 44 ". For a generalization see G. P61ya, Jour, fur Math., 



145, 1915, 233. 



440 Atti della R. Accad. Lincei, Rendiconti, 22, I, 1913, 213-5. 

 tt Sitzungsber. Akad. Wiss. Berlin, 1912, 660-70. 

 Math. Annalen, 74, 1913, 271-4. 



47 L'enseignement math., 15, 1913, 200-1. 



48 L'interme'diaire des math., 20, 1913, 177, 179. 



48 Beitrage zum Waringschen Problem, Diss., Gottingen, 1913, 74 pp. 



