CHAP. XIX] SQUAKES WHOSE SUMS BY THREES ARE SQUARES. 503 



latter value 4 and dropping the common factor x, we get 



A=a B _y-g(rn c j'-i-y 



A-g, B, C- ! , D ft f 



Using the former value and taking y = 2g, g = m{n, and multiplying A, -, 

 Dby (/ 2 +l)w 4 , we get 



A = 2wm 3 ( / 3 + 1) , B = ^(?w 4 +?i 4 ) - 2mn*( f 2 - 1) , 



C=(/ 2 -l)(m 4 +7i 4 )-4/?ww 3 , Z) = 2wX/ 2 +l). 



In his second method ( 56-60), Euler denoted the squares by v 2 , z 2 , 

 ?/ 2 , z 2 . Let a, a be two numbers for which a 2 +ar = A 2 . Let 



Av+ax\ 2 Ax+av\ 2 



\ a a / 



The first two lead to a single condition and the last two to a single one : 

 fl 2 (2/ 2 +2 2 ) =cr(V+z 2 ) +2aAvx, ' a 2 (v 2 +x"}=a 2 (if+z' 2 ) -2aAyz. 



By adding these two equations, we get z = avxf(ay). The firet of the two 

 becomes 



Q; 2 y 2 ?/ 2 a 2 ?/ 4 . 



To make rational, set y = ?/(l+s), E = ?/(A+2a 2 s/A+as 2 ). Of the 

 resulting two solutions, one is complicated, while the other (given by 

 x/y = 1) is 



y = a(A 2 -2o; 2 ), x = y = 2aaA, z = a(A 2 -2a~). 



To obtain a simpler solution in which the numbers are distinct, take two 

 numbers 6, (3 such that 6 2 +/3 2 = 2 , and set av" = pM, ay 2 = bM. Then 

 ^R = BM. But a@/(ab} must be the square of v/y; take it to be m 2 /n 2 . 

 Thus 



v _m x _AbmaBn z _am x 



y n ' y afin abn' y an y' 



Taking a = 21, a = 20, 6 = 35, = 12, we get A = 29, 5 = 37, m = 3, n = 5. 

 For the lower sign, xfy = 3/8. Hence =168, a; = 105, ?/ = 280, 2 = 60. 

 Finally, he noted the solution 



M. S. O'Riordan 33 developed the idea underlying Euler 's first solution. 

 Let S = A 2 +B 2 +C 2 +D 2 , S-A 2 = a 2 , - - -, S-D 2 --=d 2 . To obtain a number 



33 The Gentleman's Math. Companion, London, 2, No. 12, 1809, 185-7; Math. Repository 

 (ed., Leybourn), New Series, 6, II, 1835, 1-4. Reproduced in Math. Magazine, 2, 1898, 

 218-9. 



