548 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



if e = l. For e = 2, take s = 2, whence d = and the triangle is equilateral. 

 No general discussion was given. 



C. F. Kausler 17 gave a complex and inconclusive argument to show that 

 x*y* is not a cube. His first theorem is that xy and x 2 +xy+y* are not 

 both cubes; the proof rests on Euler's 8 lemma about p~-\-3q 2 a cube. 



C. F. Gauss 18 proved by descent that x 3 -\-y z -\-'z 3 = is impossible in 

 integers, using an imaginary cube root of unity. 



P. Barlow 19 gave an erroneous proof [Barlow 15 of Ch. XXVI]. 



A. M. Legendre 20 proved that the even one of x, y, z is divisible by 3 

 and then by descent that x 3 -\-y* = (2 m 3 n u) 3 is impossible, where u is not 

 divisible by 2 or 3. 



Schopis 21 undertook a proof of the impossibility of 



in integers. If the equation holds, then ?/ 3 Q = cube, Q = 2 3 , where 



Q~,2 Oxv, 



Q=~+- 

 y z y 



Solving for y, we get 





2(s 3 -l) 



Thus 122 3 3 = w 2 . The quotient of w 2 +3 by 12 must be an integer, 

 whence w = 6n+3, and 



He stated that the second member is a cube only when n = G or 1, whence 

 2 = 1, and the denominator of y would be zero. 



L. Calzolari 22 attempted to prove the equation impossible. 



L. Kronecker 23 noted that the theorem that r 3 +s 3 = l has no rational 

 solutions with rs + is equivalent to the fact that 4a 3 +276 2 = 1 has no 

 rational solutions other than a= 1, 6 = 1/3. The latter are the only 

 values of the coefficients of a cubic x s -\-ax-\-b = with rational coefficients 

 and discriminant unity. 



G. Lame 24 noted that, if x and y are relatively prime, x s -\-y 3 is the product 

 of two relatively prime factors 5, q, where 5 is D = x-\-y or 3D according as D 

 is not or is divisible by 3, and q is of the form A 2 +3B 2 . Then if a sum of 

 two cubes is a cube, we transpose the single even cube and get x 3 +y* = (2^) 3 , 



17 Nova Acta Acad. Petrop., 13, ad annos 1795-6 (1802), 245-54. 



18 Werke, II, 1863, 387-390, posthumous MS. Quoted, Nouv. Corresp. Math., 4, 1878, 136. 



19 Theory of Numbers, London, 1811, 132-140. 



20 M6m. Acad. Roy. Sc. de PInstitut de France, 6, annee 1823, 41, 49 ( = Suppl. 2 to Th<5orie 



des nombres, ed. 2, 1808). The'orie des nombres, ed. 3, 1830, art. 653, pp. 357-60; Ger- 

 man transl. by Maser, II, 348. 



21 Einige Satze aus der unbestim. Analytik, Progr. Gumbinnen, 1825. Repeated in Zeitschr. 



Math. Naturw. Unterricht, 23, 1892, 269-270. 



22 Tentative per dimostrare il teorema di Fermat . . . , Ferrara, 1855; Extract by D. 



Gambioli, Periodico di Mat., 16, 1901, 155-8. 



23 Jour, fur Math., 56, 1859, 188; Werke, I, 121. 



24 Comptes Rendus Paris, 61, 1865, 921-4, 961-5. Extract in Sphinx-Oedipe, 4, 1909, 43-4. 



