CHAPTER XXV. 



WAKING'S PROBLEM AND RELATED RESULTS.* 



WAKING'S PROBLEM. 



E. Waring 1 stated that every integer is a sum of at most 9 [positive 

 integral^ cubes, ^Iso a sum of at most 19 biquadrates, etc. Every integer N 

 of the proper form is a sum of a finite number of terms t = ax m -\-bx n -\-cx r -\- 

 (N being a multiple of 3 if = 3z 4 +6z 3 +24). Cf. MaiHei,. 14 



J. A. Euler 2 stated that, to express every positive integer as a sum of 

 positive nth powers, at least T = v+2 n 2 terms are necessary, where v 

 is the largest integer <(3/2) B . For n = 2, 3, 4, 5, 6, 7, 8, T=4, 9, 19, 37, 

 73, 143, 279 [cf . Vacca 18 ]. 



A. R. Zornow, 3 at the suggestion, of C. G. J. Jacobi, constructed a table 

 of the least number of positive cubes composing each number ^3000. 

 The number of cubes was stated to be ^8 except for 23, ^7 for numbers 

 >454, ^6 for numbers >2183. The final statement and the second for 

 239 (which requires 9 cubes) are erroneous. Corrections were made by Z. 

 Dase, who computed a table extending to 12000 and communicated it to 

 Jacobi. 4 The largest number within the limits for which 7 cubes are 

 required is 8042; for 8 cubes, 454. Jacobi considered the problem to find 

 all the decompositions of a given number into the least number of cubes. He 

 tabulated the numbers < 12000 which are sums of two cubes and those 

 which are sums of three cubes. 



G. A. Bretschneider 5 constructed at Jacobi's suggestion, a table giving 

 all the decompositions of numbers ^4100 into a sum of biquadrates, and a 

 companion table showing the numbers which equal the sum of a given 

 number of biquadrates but not fewer. For 79, 159, 239, 319, 399, 379 and 

 559, it is necessary to use 19 biquadrates; for the remaining numbers, at 

 most 18. As far as 4096 = 4 6 , he verified that 37 fifth powers are needed, 

 and 73 sixth powers. He repeated Euler's 2 statement. 



J. Liouville 6 was the first to prove that every positive integer is the sum 

 of a fixed number N of biquadrates, in fact, of at most 53. He first proved 

 that the product of any square by 6 is a sum of 12 biquadrates, in view of 



*A. J. Kempner read critically the reports in this chapter and compared them with the 

 original papers except for 2, 6, 386, 44a, 54, 60-62, 64, 69, 72, which were not accessible 

 to him. The statements concerning incorrect results in papers 6a, 13 and 17 are made 

 on his authority. 



1 Meditationes algebraicae, Cambridge, 1770, 204-5; ed. 3, 1782, 349-350. 



2 L. Euler's Opera postuma, 1, 1862, 203-4 (about 1772). 



3 Jour, fur Math., 14, 1835, 276-280. 



4 Jour, fur Math., 42, 1851, 41-69; Jacobi, Werke, VI, 322-354, and 429-431 for corrections 



of the Journal article. 



5 Jour, fur Math., 46, 1853, 1-28. 



* In his lectures at the College de France; printed in V. A. Lebesgue's Exercices d' Analyse 

 Nume"rique, Paris, 1859, 112-5. Cf. E. Maillet, Bull. Soc. Math. France, 23, 1895, 

 bottom of p. 45. 



717 



