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



E. Landau 32 proved that every integer z exceeding a fixed value is the 

 sum of at most 8 positive cubes. He proved that there exists a prime p 

 not dividing z such that Sp g ^z < I2p g and such that p 2 (p 1) is not divisible 

 by 3. Hence /3 3 =z (mod p 3 ) has positive integral solutions /3<p 3 . In 

 2=j8 8 +pW, set M=6p 6 +Mi. Then 



7p 9 <p z M<12p*, p 6 <Mi<6p*. 



By the paper of Wieferich, 30 we can find an integer 7, 0^7<96, such that 

 Mi 7 3 = 6m, where m = xl+x 2 2 -\-xl. For z sufficiently large, m<p 6 , so 

 that Q^=Xi<p 3 , and 



1=1 



A. Wieferich 33 proved that 7V 5 =59, A^ 7 ^3806. He gave a table showing 

 the least number of fifth powers required to represent each number !,, 

 3011. 



D. Hilbert 34 proved Waring's assertion that every positive integer z 

 is the sum of at most N m positive wth powers, where N m is a finite number, 

 not determined, depending upon m but not upon z. He first proved, by 

 use of a five-fold integral (a 25-fold integral in the first paper,) the lemma 

 (stated by Hurwitz, 28 who was unable to prove it) that there exists for every 

 m (and r = 5) an identity in the z's 



. +a rh x T Y m , 



where the a,/, are integers and the p h are positive rational numbers. It is a 

 simple step to prove Waring's theorem for powers whose exponents are 

 2 fc , k ^ 2. The case of any exponent is derived from this by an elemen- 

 tary, but long, discussion (not using calculus). 



F. Hausdorff 35 proved Hilbert's lemma by use of integrals involving 

 exponentials, the method being more suitable for computing the a's and 

 p's. 



E. Stridsberg 36 proved easily that Waring's theorem for /xth powers 

 would follow if it were shown that, if B is any real number, every positive 

 integer ^B can be written as Sp x P, where the P's are integers ^0 

 and p x is a positive rational number depending only on ju. He noted 

 that HausdorfFs elegant modification of Hilbert's proof can be reduced to 

 an elementary study of binomial coefficients. Using symbolic powers of h, 

 let /i 2M denote (2ju)!//*! for all even integers 2^^0, and /i 2M+1 = for all odd 

 integers 2/*+l ^1. A theorem of Hausdorff's becomes the simple one that, 

 if f(x) is any polynomial which is never negative for a real value of x, then 



32 Math. Annalen, 66, 1909, 102-5; Landau, Handbuch der Lehre von der Verteilung der 



Primzahlen, 1, 1909, 555-9. Cf. Landau. 39 

 83 Math. Annalen, 67, 1909, 61-75. 



" Gottingen Nachr., 1909, 17-36; Math. Ann., 67, 1909, 281-300. 

 46 Math. Annalen, 67, 1909, 301-5. Cf. Hurwitz. 44 

 18 Arkiv for Mat., Astr., Fysik, 6, 1910-11, No. 32, No. 39. French re'sume' in Math. Annalen, 



72, 1912, 145-152. 



