CHAP, xiii] ax*-{-bif-\-cz 2 = Q. 423 



arbitrary integers, let a 2 , /3 2 , y 2 be the largest squares dividing be, ac, ab, 

 respectively, and set aa = pyA, (3b = ayB, yc = a@C; then the former equa- 

 tion is solvable if and only if AX 2 +BX 2 +CZ 2 = Q is solvable, and the latter 

 falls under the above theorem since A, B, C are relatively prime in pairs and 

 have no square factors. For, bc/a. 2 = BC is an integer without square 

 factor, so that B, C are relatively prime and without square factors. 



E. F. A. Minding 117 considered x 2 = Ay 2 +Bz 2 , where A, B are without 

 square factors. Let / be the g.c.d. of A = af and B = bf. The equation is 

 solvable if and only if A, B, ab are quadratic residues of B, A, f, re- 

 spectively. 



A. Genocchi 118 treated the equation az 2 +bx 2 = (a+b}y 2 , equivalent to (1), 

 by the methods of Lagrange and Paoli 91 of Ch. XII. 



G. L. Dirichlet 119 treated ax 2 +by 2 +cz 2 = Q, where a, b, c are relatively 

 prime in pairs. If u, v, w are given relatively prime solutions, we can deduce 

 all solutions. Since au, bv, cw are relatively prime and au, for example, is 

 even, we can find integers I (even), m and n such that aul-\-bvm+cwn=l. 

 Set al 2 +bm 2 +cn 2 = h. Then u' = 2lhu, v' = 2mhv, w' = 2n hw are solu- 

 tions, congruent to u, v, w, respectively, modulo 2. Hence, in 



" 



u", v", w" are integers. If x, y, z are any integers, 



(6) t = au'x+bv'y+cw'z, t' = aux+bvy+cwz, t" =u"x+v"y+w"z 



are integers and t=t' (mod 2). It is shown that, conversely, if t, t r , t 

 are any integers for which t t' is even, 



2x = ut+u't' - 2bcu"t", 2y = vt+v't' - 2cav"t", 

 2z = wt+w't'-2abw"t", 



so that x, y, z are integers. Multiply the latter equations by ax, by, cz, 

 add, and use (6). We get 



ax 2 + by 2 + cz 2 = it' - abet" 2 . 



Hence if x, y, z are solutions of the initial equation, then t, t', t", defined by 

 (6), are integers for which t=t' (mod 2) and tt'=dbct" 2 . Conversely, if 

 t, t f , t" are integers satisfying the last two conditions, the values of x, y, z 

 given by (6') are integral solutions. Further, by use of the above relations 

 he proved the following extension of Legendre's 114 theorem: If no two of 

 a, b, c have a common factor and are not zero, ax 2 -\-by 2 -\-cz 2 = Q is solvable 

 in relatively prime integers if and only if be, ca, ab are quadratic 

 residues of a, b, c, respectively, and the latter are not all of the same sign; 

 further, if bc=A 2 (mod a), ca=B 2 (mod 6), ab^C 2 (mod c), there 

 exist relatively prime solutions for which 



Az =by (mod a), Bx=cz (mod b), Cy=ax (mod c). 



117 Anfangsgriinde der Hoheren Arith., 1832, 84. 



118 Annali di Sc. Mat, e Fis., 6, 1855, 186-194, 348. 



119 Zahlentheorie, 156-7, 1863; ed. 2, 1871; ed. 3, 1879; ed. 4, 1894. 



