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



x = w, y = u z z and x = u?, y = uz lie on the surface (10) with u = l. Each of 

 these generators meets the line 



x = az-{-b, y pz-\-q 

 if 



co b_ q co 2 b_ q 



a co 2 p j a co p' 



whence p = b, q=(l+b-\-b 2 )/a, and the ^-coordinates of the points of inter- 

 section are respectively 



co 6 co 2 b 



Zi = ~ , Z2=- . 



a a 



The third root of (az+b) 3 +(pz+q} 3 = z 3 +l is 



_ 



Then also x and y are rational in a, b. To obtain simpler formulas, replace 



a by I/a, b by b/a. Then 



(12) sx = r(a+2fy-l, sy = r 2 -a-2b, sz = r 2 -a+b, 



where r = a 2 +a&-f& 2 , s = a 3 b 3 1. Passing to the homogeneous equation 

 (10) and changing 6 to 2b, a to a b, we get (11) with x, y, z, u replaced by 



2> y> x t -~u. 



Several 63 expressed 8+27 and 1+8 as sums of two new rational cubes. 



G. Korneck 64 stated that all integral solutions are obtained by taking 

 positive and negative integers m, t, f in 



x = 6m 3 tf+t(tm)r+3t(t=Fm)f 2 , y = 6mHf-t(tm)r-3t(t : =Fm)f 2 , 



z= -Qt 3 mf+m(mt)r+3m(m^f)f 2 , u = 6t 3 mf+ 

 where r = m 4 -\-mH 2 -\-t i . 



E. Catalan 65 noted that (4) is satisfied identically by 



S. Re*alis 66 proposed a problem which was solved by P. Sondat; 67 if a, 

 7, 5 is one set of solutions of x 3 -\-y s -\-u s -}-vP = Q, another set is 



u = aA B, v = (3AB, x = ^A-\-B, y = dA-\-B, 



A = a + /3+7 + 5, = a 2 +/3 2 -7 2 -5 2 . 



The new set yields similarly the given set, apart from a common factor. 

 G. Brunei 68 treated, for n an odd prime, the equation 



'i 2/2 2/n-i 



(13) 







2/2 2/3 



Math. Quest. Educ. Times, 16, 1872, 95-6; 17, 1872, 84. 



64 Auflosung x 3 +y*+z 3 = u 3 in ganzen Z., Progr. Kempen, 1873. 



88 Nouv. Corresp. Math., 4, 1878, 352-4, 371-3. Cf. Catalan. 123 



" Nouv. Ann. Math., (2), 17, 1878, 526; Nouv. Corresp. Math., 4, 1878, 350. 



87 Nouv. Ann. Math., (2), 18, 1870, 378. 



88 M6m. Soc. Sc. Phys. et Nat. de Bordeaux, (3), 2, 1886, 129-141. 



