570 HISTORY OF THE THEORY OF NUMBERS, [CHAP, xxi 



v 



Thus x = p 2 +2nqr, y = 2pq-\-nr 2 , = 2pr-fg 2 , which holds il p = 2a 2 , r = fc 2 , 

 q = 2ab. The product of the three factors is the square of p 3 +ng 3 -fnV 3 

 Snpqr. 



J. L. Lagrange 161 proved that r 3 As z = q z for 



(1) r = 4:t(t*-Ay?}, s= -u(8P+Au 3 ), q 

 He took a cube root a of unity and set 



Thus 



T = t-+2Aux, U = Ax 2 +2tu, X = 



Then the factor r as A/ ~A. of the given cubic function will be of the form 

 p 2 if r = T, s = - U, X = 0. Substituting the value x = - u-/(2t) from X = 

 into the first two conditions, we get 



Au* 



T' "IF" 



In the product P = t 3J rAu 3 3Atux+A-x* of the expressions p in which a 

 takes its three values, we insert the above value of x and obtain q. To 

 avoid fractions multiply r and s by 4 2 , and q by St 3 . 



Euler 162 noted that this product P may be made equal to any power. 

 Lagrange 163 extended the method from a 3 = 1 to a 3 aor -\-ba- c = 0, with 

 the roots a lf 2 , a 3 - Then 





F(x, y, z)= 



+ (b~-2ac)xz"+af+acy-z+bcyz-+c-z* 

 is such that its product by F(XI, yi, 2i) is F(X, Y, Z), where 



In particular, the square of F(x, y, z) is F(X, F, Z), where 



X = x*+2cyz+acz'>, Y = 2xy-2byz+(c-ab)z-, 



Z = 2xz+if+2ayz+(a--b)z~. 

 We may make Z = by choice of x rational in y, z. Hence 



has solutions involving the parameters y, z, with V = F(x, y, z}. The same 

 method leads to solutions of F(X, Y, Z) = V m . 



A. M. Legendre 164 made Z = by taking y = (u ct)z, 2x=(b u' 1 }z. 

 Then replacing u by u/v, we see that X, Y, V are proportional to 



X = 

 F = it 6 - 



161 M6m. Acad. R. Sc. Berlin, 23, ann6e 1767, 1769; Oeuvres, II, 532. 



162 Opera postuma, 1, 1862, 571-3; letter to Lagrange, Jan., 1770, Oeuvres, XIV, 216. 



193 Addition IX to Euler's Algebra, 2, 1774, 644-9 [misprint of sign in X, 92]. Oeuvres de 



Lagrange, VII, 170-9. Euler's Opera Omnia, (1), I, 643-50. 

 164 The"orie des nombres, ed. 3, II, 1830, 465, p. 139. German transl. by Mascr, 2, 1893, 133. 



