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



Take A +B = 90. Then x*+y 2 = a? if z = 2asin2A. Take cot|A=r/s, 

 a=(r 2 +s 2 ) 3 . 



C. L. A. Kunze 14 set x = 2mn, y = m?n?, z = m 2 a/bn~bla. Then 



Take a, fe to be legs of a rational right triangle with hypotenuse h, and 

 set n = mh/(2ty. Then y--\-z~ = Wn 4 /a 2 . Multiplying the resulting x, y, z 

 by 4afe 2 /?ft 2 , we get 



The last solution was obtained also by taking 



x = 2mn, y = mri i m, z = nm-n. 

 Then the first two conditions are satisfied, while 



?/ 2 +2 2 = m 2 n 2 (w 2 -fn 2 -4) +m 2 +n 2 = D 



if m?+n z = 4i. Take m = 2alh, n = 2b/h, 2 +b 2 = /i 2 , and multiply x, y, z 

 by /i 3 /2. We get the former solution. 



Judge Scott 15 took x 2 +y 2 =(ym} 2 , x 2 +z 2 = (zn) 2 , which determine 

 y, z. Take mfs = (p 2 (? 2 )/(;p 2 +<? 2 ), n/s = 2pql(p' 2 -{-q-), whence m 2 -fn 2 = s 2 . 

 Then 2/ 2 +2 2 =D if s 2 o; 4 4m 2 n 2 a; 2 +m 2 nV= D =s 2 a; 4 , say, whence x = s/2. 

 Take s = 16pg(p 4 g 4 ). We get Euler's 6 answer. The latter was ob- 

 tained also by A. Martin (ibid.), who satisfied u?y 2 = w' 2 z* by taking 

 U = a(r 2 -s 2 ) ) 7/ = 6(r 2 -s 2 ), w = a(r 2 +s 2 )-26rs, z = 6(r 2 -{-s 2 )-2ars. Then 

 u 2 7/ 2 =D=^ 2 if a = p 2 +g 2 , b = 2pq. There remains the condition 

 2/ 2 +z 2 =CK Divide by 4r 2 s 2 and take ?n=(r 2 +s 2 )/(rs). Then a quartic 

 in p is to be a square, say (p 2 mpq q 2 ) 2 , whence p/q = 4fm. 



C. Chabanel 16 used the devices of Diophantus for a similar problem. 

 Set 



Then z 2 +s 2 =( T A+7i0 2 , ^+2 2 = (5A+5!0 2 . Since 



which is a square for t = zj(a 4 ^) since 7 2 + 5 2 = (cr+/3 2 ) 2 and 



7? + ^ = ( 4 -/3 4 ) 2 . 

 Multiplying the initial x, y, z by 2(cr+/3 2 ) V2/3, we get 



for 



Y, r = 2^[( 2 +/3 2 ) 2 T4(a 2 -^ 2 ) 2 ], p = ( 2 +/3 2 ) 3 , 

 where the upper signs give X, Y. For a = 2, /3 = 1, we get Halcke's 1 solution. 



14 Ueber einige Aufg. Dioph. analysis, Weimar, 1802, pp. 7-9. 

 16 Math. Quest. Educ. Times, 17, 1872, 82-3. Cf. Martin 20 . 

 16 Nouv. Ann. Math., (2), 13, 1874, 289-292. 



