CHAP. XXV] SUMS OF UNLIKE POWEKS. 725 



(mod 72). He reduced the limit for N 6 to 478, that for N 6 to 58 and gave 

 a simpler proof that N 4 ^37. For & = 2744, it is shown by elementary 

 methods that every number = k (mod 2k} is a sum of 7 or fewer positive 

 cubes; hence if Ci(x) denotes the number of positive integers ^=x which are 

 decomposable into 7 or fewer positive cubes, 



(2) < - ' ^1 for all sufficiently large x. 



K X 



His transcendental methods enabled him to replace If (2k) by 13/72. He 50 

 later gave a direct elementary proof of the last result (2) for A; = 4096 by 

 noting that the integers ku, where u is positive and odd, can be decom- 

 posed into 7 positive cubes all of whose 7 bases exceed any assigned positive 

 number g for every u exceeding a limit depending upon g. 



E. Stridsberg 51 gave a brief elementary proof of Hurwitz's lemma [Hil- 

 bert 34 ] without the use of integrals (Remak, 43 Frobenius 45 ) or the gamma 

 function. The proof is admitted to be otherwise essentially the same as 

 his 36 former proof. 



G. H. Hardy and S. Ramanujan 52 proved that the logarithm of the 

 number of ways n is a sum of rth powers of positive integers (rearrangements 

 of the same powers not being counted as distinct) is asymptotic to 



r/(r+l) 



where f denotes the Riemann zeta function, and F the gamma function. 



Hardy and J. E. Littlewood 52 " made use of the theory of analytic func- 

 tions (cf. Ch. Ill 221 ) to prove that every positive integer, which exceeds a 

 certain number depending on k alone, is a sum of at most k-2 k ~ lj r\ posi- 

 tive kth powers; for example, a sum of at most 33 biquadrates. The 

 transcendental method leads not only to a proof of the existence of repre- 

 sentations, but also to asymptotic formulas for their number. They since 

 communicated to the author the improved result that at most (k 2)2*~ 1 +5 

 positive kih powers are necessary; this gives 9 cubes, 21 biquadrates, 53 

 fifth powers, 133 sixth powers, etc. 



NUMBERS EXPRESSIBLE AS SUMS OF UNLIKE POWERS. 



D. Andre 53 proved that every even integer is the sum of a cube =1=0 and 

 three squares (since every 8n+3 is a OS). In general, if s is odd, every 

 even integer >7 S is the sum of an sth power +0 and three squares each 4=0. 



G. de Rocquigny 54 noted that every integer except 1, 2, 3, 4, 5, 7, 8, 

 10, 11, 18 is a sum of three cubes and three squares. He 55 stated many 



60 Math. Annalen, 74, 1913, 511-4. 



61 Arkiv for Mat., Astr., Fysik, 11, 1916-7, No. 25, pp. 35-9. His second paper with the 



same title, ibid., 13, 1919, No. 25, deals at length (pp. 31-70) with definite and semi- 

 definite polynomials in x and incidentally with their occurrence in the literature on 

 Waring's problem. 



62 Proc. London Math. Soc., (2), 16, 1917, 130. 

 62 Quar. Jour. Math., 48, 1919, 272 seq. 



" Nouv. Ann. Math., (2), 10, 1871, 185-7. 



64 Travaux Sc. de 1'Univ. Rennes, 3, 1904, 42. 



L'interme'diaire des math., 10, 1903, 109, 212; 11, 1904, 31, 56, 81, 99, 149, 171, 214. 



