CHAP. XIII] az 2 +fa/ 2 +C2 2 = 0. 421 



M be the g.c.d. of V and x+y and set V = MN, x-\-y = Mp. Then z 2 = pa, 

 x y = N(7, where a- is an integer. If I is the g.c.d. of p and a, we have 

 p = lm 2 , <j = In 2 , whence 



z = Imn, x = l(Mm 2 +Nn 2 )/2, y = l(Mm 2 -Nn 2 )/2. 



We may set I = 2, since we may multiply x, y, z by 2/L 



L. Euler 111 stated that the general solution of ax 2 +(3y 2 = yz 2 is given by 



if one solution af 2 -{-^g 2 = yh 2 is given; the solution by taking n = l is not 

 general. Again, by taking x=fp+(3gq, y = gp afq, we get ax 2 +^ 2 = 7/i 2 J R, 

 where 72 = p 2 +aj8g 2 is the square of r 2 +o;/3s 2 for p = r 2 a(3s 2 , q = 2rs. Again, 

 if we multiply the initial equation by h 2 and af 2 -\-pg 2 = yh 2 by z 2 and sub- 

 tract, we get 



a(hx+fz) 



gzhy hx-fz 



Set each fraction equal to pfq and equate the two values of z; we get y/x. 

 To obtain another solution, set F = apq 2 p 2 , G = 2pq, H = a(3q 2 -\-p 2 , whence 

 H 2 = F 2 +a(3G 2 . Multiply the latter by yh z = af*+pg*. The product of the 

 right members leads to the solution 



z = hH, x=fF+(3gG, y = gF-afG. 



A necessary condition for fx 2 +gy 2 = hz 2 is that fg be a quadratic residue 

 of h. 



Euler 111 " made ax 2 +cy z a square by use of 



c) 2 . 



Euler 112 considered the rational solutions of 

 (5) fx 2 +gif = hz 2 . 



If, for / and g fixed, the equation is solvable when h = hi, h z and h 3 , then it 

 is solvable when h=hih z hz. He stated (p. 558) the elegant empirical 

 theorem that if (5) is solvable when h = hi it is solvable also when 

 h = hinfg, provided the latter is a prime. 113 



If (5) be solvable, then (p. 566) fg is a quadratic residue of h. For, 

 since x, y may be taken relatively prime, we can determine p, q so that 

 pyqx = l. Then 



( fx 2 +gy 2 ) ( fp'+gq 2 } = t 2 +fg (t =fpx+gqy) 

 is divisible by h. 



111 Opera postuma, 1, 1862, 205-211 (about 1769-1771). 



1110 Algebra, St. Petersburg, 2, 1770, 181-7; Lyon, 2, 1774, pp. 219-26; Opera Omnia, 

 (1), I, 425-9. Cf. Euler 6 and Lagrange 63 of Ch. XX. 



lu Opusc. anal., I, 1783 (1772), 211; Comm. Arith., I, 556-569. 



113 A. M. Legendre, Me"m. Acad. Sc. Paris, 1785, 523, stated that this theorem is true, but 

 omitted the proof (not easy) as it was necessary to separate cases. He stated the 

 generalization: If fx 2 gy 2 =hz z is solvable then /z 2 <7?/ 2 = c2 2 is solvable if c = h-}-fgn 

 is a prime and if n is such that the two members of the quadratic equation are congruent 

 modulo 8. 



