CHAP, xxv] WAKING'S PROBLEM. 721 



E. Maillet 26 proposed the following generalization of Waring's problem: 

 Can k be taken sufficiently large that there shall be integral solutions of 



where n\, , n a have given values, and Ni, - - , N a any values satisfying 

 suitable conditions? For a = 2, Wi = 2, n 2 = l, & = 4, there is always a 

 solution (Cauchy, Ch. VIII, p. 284) if Ni is odd and Nz is odd and 



E. Maillet 27 proved Waring's theorem for eighth powers, but gave no 

 explicit limit for N*. He proved in an elementary way that there is an 

 infinitude of numbers each not a sum of n or fewer nth powers. 



A. Hurwitz 28 proved that every integer is the sum of at most 



37(6-4+60-12+48+6. 8)+5039 = 36119 

 8th powers, in view of A^ 4 ^37 and the identity 



+ 



48 



In general, if there exists an identity (in a, 6, c, d) 



p^+V+ct+d^P&ia+Pib+yiC+dW, 



i=l 



where p, p\ f , Pi are positive integers and ai, -, d r are integers, then 



N Zn ^Nn(pl+'"+PrHp-l, 



so that N Zn would be finite if N n is. He proved by use of the gamma 

 function that there is an infinitude of positive integers each not the sum of 

 n or fewer nth powers. 



J. Schur 29 found the identity which proves Nio finite : 



12 



i 



A 

 48 



A. Wieferich 30 proved that N 3 ^9 [except fora limited set of integers 

 arising from a case 31 overlooked^]. The proof consists in showing that any 

 positive integer is the sum of three cubes together with k = 6a 3 +6am, where 

 0<A and m =x\ -\-xl+xl <A 2 . For, by Maillet, 11 k is then a sum of 6 

 positive cubes. 



28 L'intermddiaire des math., 15, 1908, 196, and Maillet." 



27 Bull. Soc. Math, de France, 36, 1908, 69-77; Comptes Rendus Paris, 145, 1907, 1399. 



28 Math. Annalen, 65, 1908, 424-7. 



29 Math. Annalen, 66, 1909, 105 (in a paper published by Landau.) 

 80 Math. Annalen, 66, 1909, 95-101. 



31 The case ?=4 in 10648< (0.4)5 2p/ - e . Attention was called to this gap in the proof by P. 

 Bachmann, Niedere Zahlentheorie, 2, 1910, 344, who indicated in his Zusatze, pp. 477-8, 

 a long method of treating the omitted case, but himself made certain errors. The latter 

 were incorporated in the unsuccessful attempt by E. Lejneek (Math. Ann., 70, 1911, 

 454-6) to fill the gap. The gap in Wieferich's proof was filled by Kempner. 42 



47 



