728 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XXV 



Again, if p= -1-5/480, 



= 30(p+2) 4 +2(p-2) 4 -20(p+l) 4 -12(p+3) 4 . 



These two quartic forms repeat under multiplication. 



Libri 65 proved that any positive rational number ra equals the sum of 

 four positive rational cubes. In the identity 



we can reduce the right member to a sum of four positive cubes. In 



take a = (ra+6<? 3 )/(6g 2 ), 6 = ra/(6g 2 ). Then the sum of the first and third 

 terms in (2) is a sum a 3 -j-/3 3 of two positive cubes if (ra+6<f) 3 >2m 3 , where 



_m+6g 3 {(ra+6g 3 ) 3 -2ra 3 } 



' 



Now use (3) for a = a, b = m/(Qq y ). Then a 3 {m/(6g 2 )} 3 is a sum of two 

 cubes each positive if 



which implies the preceding hie quality and can be satisfied. Formula 

 (1) is here repeated. It is stated that 3z 4 +2/ 4 2 4 3w 4 represents all 

 rational numbers. 



P. Tardy 66 gave the generalization to n factors of 4db = (a+fr) 2 (a 6) 2 

 and 



This formula had been given by C. F. Gauss. 67 



E. Rebout 68 noted that, hi this formula, also 24a6c is a cube if a = 3, 

 6 = 4, c = 6. 



V. A. Lebesgue 69 remarked that every positive rational number is a sum 

 of four positive rational cubes : 



(4) n 



where ra 3 is a rational cube lying between n/Q and w/12, while 



<z = l+6m 3 /n, 6 = 2-3/(a 3 +l), c = 2-3/(6 3 +l). 



65 Jour, fur Math., 9, 1832, 288-292; Mem. pre'sente's pars divers Savants Acad. R. Sc. 



1'Institut de France (Math. Phys.), 5, 1838, 71-5. In Comptes Rendus Paris, 10, 1840, 

 313, Libri stated he had proved the theorem in his book, *Memoires de Math, et de 

 Phys., Florence, 1829, 152-168. 



66 Annali di Sc. Mat. Fis., 2, 1851, 287; cf. Nouv. Ann. Math., 2, 1843, 454. Cf. Boutin." 

 87 Werke, II, 1863, 387. Cf. H. Brocard, Nouv. Corresp. Math., 4, 1878, 136-8. 



68 Nouv. Ann. Math., (2), 16, 1877, 272-3. 

 "Exercices d'analyse numdrique, Paris, 1859, 147-151. 



