CHAP. XVIII] FOUR QUADRATIC FUNCTIONS MADE SQUARES. 495 



The final expression is a square if and only if 



pq(p 4 q*) -rs(r 4 s 4 ) = D. 

 By special assumptions, Euler was led to the values 



Then 



r 2 +s 2 =(p+#>, w=cr-f Ga/3+13/3 2 , 



r-s= (3/3+a)(3/3-a), p-q = s. 



The condition is thus (a +3/3) (a +7/3) (a 2 +0 2 ) = D, which is treated by the 

 usual methods. From a = 2, = 1 and a= 17, p = 7, we get the solutions 



l^) 2 5-29 2 13 2 -53 2 37-13 2 -53 2 



' 8-9 2 ' 32-11 2 ' 3-4-7-59 2 ' 3-7-4 2 -5 2 -19 2 ' 



Euler 20 made x+yx z , x+yy- all squares. Replace x, y by xjz, y/z. 

 Then (x+y}zx z , (x-{-y}zy~ are to be squares. This will be the case if 



The final equality holds if 



A=ac+bd, B = ad bc, C = ad+bc, D = ac bd. 



The first two conditions hold if x = Af, y = Cg, 2B = A/ 2 , 2D = Cg\ By the 

 latter, 



cf b2c-dg' 2 

 which are equal if 



Hence the problem will be solved if we make B=D. Set gr = l. Since 

 R = G for / = 1 (which makes x = y),we take /= 1+t. Then R is the square 

 of 5+2+13Z 2 /5 if 2 = 60/11. Dropping the common factor 13 in x, y, we 

 get 



5 37 2 -61 2 

 3 = 4.11-71, ?/ = 4- 37-61, g= 2 . 49 . 31 - 



20 M6m. Acad. Sc. St. Ptersbourg, 11, 1830 (1780), 46; Comm. Arith., II, 447. 



