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



E. Landau, A. Boutin, P. Tannery, and A. S. Werebmsow 149 considered 

 x 3 +3x 2 y+6xif+2y 3 = l or z\ 



P. von Schaewen 150 treated Ax 3 +Bx 2 y+Cxy 2 +Dy 3 = z 3 . If A = a 3 , 5 = 0, 



we have 



(z-ax)(z 2 +axz+a 2 x 2 } =y z (Cx+Dy), 



which is satisfied if m(z-ax)=ny, n(z 2 -\ ---- )=m(Cx-)rDy)y. Eliminating 

 z, we get 



- = - - {Cm 2 -3an 2 E*}, E = C 2 m*+12a 2 Dm 3 n-6aCm 2 n 2 -3aW. 

 y Qa~mn 



We can always make E a square. Next, if A=a 3 , 5=1=0, we replace 

 ax -\-Byl r (3a 2 ) by xi and y by 3a 2 ?/i and are led to the first case. Finally, 

 if neither A nor D is a cube, but x = p, y = q, z = r is a known solution, 

 set qx = py+s to obtain a cubic in which the coefficient of y* is r 3 . For 

 Fermat's example, x*+2x 2 y+3xy 2 +y* = z 3 , set X = x+y, x=Y. Then 



X 3 -XY 2 +Y 3 = z 3 , # = ra 4 +12ra 3 n+6m 2 7i 2 -3n 4 . 



Many solutions are found: (x, y, z) = (l, 1, 1), (3, 7, 1), (1, 2, 1), 

 (6, 13, 5), etc., whereas Fermat's method gave the primitive solution 

 3=19, 2/=-45. 



J. von Sz. Nagy 151 noted that a principle of Poincare's 15 of Ch. XXIII 

 enables us to transform the cubic curve f^a 3 x 3 +pxy 2 +qy 3 z 3 = without 

 double points, treated by von Schaewen, by the birational transformation 



into the quartic curve p 2 m 4 +12a 2 qm 3 n6apm 2 n z 3a~n 4 r-m 2 = Q, and con- 

 versely the last into /=0 by 

 m = y 2 , n = y(z ax} ) 



To pass to the non-homogeneous form, use x/y, zfy, nlm, rjin. 

 E. Haentzschel, 152 starting from a given solution x = h, y = k, of 



derived a second solution by applying the substitution 



gvng 



y 3 



where C 2 and C 3 are the quadratic and cubic covariants of /(x), and choosing 

 t so that 3C 2 (/i)Z+C 3 (/i) =0. We may begin with the identity 



where D is the discriminant of/, set v= C s /f, y 2 = 4s 3 +Z>; then 



"" L'interm6diaire des math., 8, 1901, 147, 309; 9, 1902, 111, 283; 10, 1903, 108; 13, 1906, 

 196-7. 



160 Jahresbericht d. Deutschen Math.-Vereinigung, 18, 1909, 7-14. 



161 Ibid., 401-2. 



M Ibid., 22, 1913, 319-29. 



