CHAP. XXII] aZ 4 + fy/ 4 = C2 2 . 631 



A. Desboves 77 employed the identity 



(2) ( 



and that obtained by changing x to x 2 and y to y 4 , to show that 



(3) x*+ay*=z* 



is solvable in integers if a is of the form (2x-\-y}x-y or 2x 2 +y*. By changing 

 x to x-\-y in (2) and making other simple transformations, he 78 proved 

 that (3) is solvable in integers if a= x*(x*+y-), y 2 x 4 , x(x+l}, 

 y(y2x 2 ), x 2 (2x+y*), y 4 -2x*, -2xy(x 2 -y*)(x 4 +y 4 -6xY) with z=D in 

 the last case; and, by other identities, if a 8(x s +y 8 ), z(x 2 +4), z 8 4. 

 If (1) has solutions x, y, z, then Fermat's method conveniently applied leads 

 to the new solution 79 



(4) X=x(4a*x 8 -3c 2 z*), y=?/(46 2 t/ 8 -3c 2 2 4 ), Z = z[4c 4 2 8 -3(ao; 4 -&i/ 4 ) 4 ], 



of different type from Lagrange's solution when a = c = 1 . For the examples 

 of Lucas 73 not under Lagrange's case and for which (4) do not give all solu- 

 tions, we have (a-}-b)c a square, say v 2 . Using fractional values, we may 

 set y = l. Then ac(x* l)+t> 2 = c 2 2 2 . Setting x=(t+l)/(t 1), we get an 

 equation for which Fermat's method is applicable. If x, y, z is & solution 

 of (1), then 80 



cz 2 , 



is a solution of x^+abc^ z 2 . The latter becomes x*+u(v 2 u~)y*=z 2 for 

 ac = u, (a+b)c = v 2 . Hence (1) is solvable if a = c l, b = u(v 2 u), as shown 

 also by the identity 



{ f\ | ( O/i / ^_ 2 A 4 ^| _ n I ( >ji2 __ 'i/i i x i) 1 4 . f /ji4 ,_, A_ii2i I LJ, / ) 1 1^\^ 



\ t/ / l^Cv ^ y I tv V v \JL j \^U ) ' It/ ri Cv I Tl Cvt/ I * 



E. Lucas 81 obtained from one solution of Xx 4 +M2/ 4= =(X+M)2 2 the two 

 solutions 



X = 4/jipn 2 x 2 y 2 z 2 mV, Y = 4\pm 2 x 2 y 2 z~ n~v 2 , 



where p = X+ju, y = Xrc 4 /*?/ 4 , m=4XjUxVdbp 2 z 4 2pxyzv, n = v 2 4\px 2 y 2 z 2 . 

 Since the proposed equation is satisfied if x = y = z = 1, we obtain two new 

 solutions. Thus 3x 4 2y* = z 2 has the solutions 



33, 13, 1871; 28577, 8843, 1410140689. 

 If (1) has the solution (XQ, yo, ZQ), it may be written in the form 



He stated that his formulas above solve completely twenty equations of 



"Comptes Rendus Paris, 87, 1878, 159-161. Reproduced, with pp. 321-2, 522, 598, in 

 Sphinx-Oedipe, 4, 1909, 163-8. 



78 Comptes Rendus Paris, 87, 1878, 321-2. 



79 Ibid., 522; correction, 599. Reproduced in Desboves' Questions d'algebre, ed. 4, 1892. 



Cf. Desboves. 123 



80 Ibid., 598. 



ft Nouv. Ann. Math., (2), 18, 1879, 67-74. In Lucas' expression for Z the coefficient 4 of 

 the final term should be 16. If we adopt his change of signs in m, we must alter a sign 

 in his Z. 



