528 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xix 



equal the square of 



if m :n = s 2 +r 2 : 2s 2 -2r 2 . 



To make a = x 2 +y 2 +s, (3 = x-+z 2 +s, 7 = y" +Z-+S squares, where 

 s = xy+xz+yz, S. Ryley 149 took y = l, z = 3. Then 



a=D, )8 = a: 2 H-4a;+12 = (x+n) 2 

 if z = (12-?i 2 )/(2n-4), and 7 (2^-4) 2 = (4-14?i) 2 if w=-16, whence 



x :y :2 = 61 : 9 : 27. 



J. Cunliffe took x = 3z, y = n z. Then 7 = (n+s) 2 . Make /3 = a 2 , by choice 

 ofn. Thenl6z 2 a = a 4 -10a 2 2 2 +153s 4 =Difa = 192/3. Ortakea = (ni-3z) 2 , 

 2 = r 2 -l, whence w = 2(3r+l). Then/3=D if r = 5/3, whence 



x :y :z = 4 : 32 : 12. 



"Limenus" took a = tf, = b 2 , <y = c 2 . Then z 2 +c 2 = 2/ 2 +& 2 = z 2 +a 2 . Hence 

 take a number (ra 2 +w?)(n 2 +n?)(g 2 +<?l) which is a sum of two squares in 

 three ways, whence 



while y (or z} is the similar expression with only the second (or first) term 

 negative. Set v = m/mi, r = n/n^ s = q/qi. Then (x+y-\-z}--\-x* = a~-\-'b- be- 

 comes/t; 2 4(r+s)y=/+4rs+4, where /=(r 2 l)(s 2 1). Thus the square 

 of fv 2rs 2s is known; equate the root to/+2rs+2+C and take C= 2 

 to cancel the terms in s 4 , s 3 . Hence 2rs= I, v=(2r~ 3r l)/(2r 2 +r 1). 

 Take q n\, q t = 2n, m = 2n 2 3nni n 2 l . Then 



x = 4?z 4 +nt n*n\ t y = 4n 4 



The least positive numbers found are 19, 13, 2. 



To make ?+y 2 +?+2r)-2tt+2r)t, etc., squares, W. Wright 150 put 

 =x+y, rj = x+z, = y+z and noted that the problem is reduced to the 

 preceding one, for which he took y = px, z = 3x, and found p so that 

 p 2 +4p+12=(p-r) 2 ; finally, 4p+ 13 = D if r = 16. Others equated the first 

 function (^+rj+^-^ to (^-f-^) 2 , whence ^ = 2^-27?, or to (2-f/2) 2 , 

 whence ^ = r/+f/2. Then the difference of the other two initial functions 

 factors. 



J. Cunliffe 151 made x*+y*+a(x+y), x-+z z +b(x+z}, y~+z z +c(y+z') 

 squares by taking x = rv, y = sv, z = tv, where r 2 +s 2 = e 2 , r 2 +^' 2 =/ 2 , s 2 +^ 2 = ^ 2 . 

 Take m = a(r+s)/e 2 , n = b(r-\-i)/f' 2 , p = c(s-\-i)/g~. Then the quotients of the 

 initial functions by e 2 ,/ 2 , g 2 are v* -\-rnv, v z -\-nv, v 2 -{-pv, which are made squares 

 (Cunliffe 1 of Ch. XVIII). 



D. S. Hart 152 equated the same initial functions to the squares of x-\-y, 

 x+z, y+z. Then a(x+y)=2xy, etc., determine x, y, z rationally in terms 

 of a, b, c. 



149 The Gentleman's Math. Companion, London, 2, No. 9, 1806, 31-35. 



160 Ibid., 5, No. 29, 1826, 502-6. 



161 Ibid., 3, No. 14, 1811, 300-2. Same by J. Matteson, The Analyst, Des Moines, 2, 1875, 



46-9. 

 Math. Quest. Educ. Times, 17, 1872, 37. 



