CHAP, xix] THREE SQUARES WHOSE DIFFERENCES ARE SQUARES. 505 



Take 



b = a cos A-\-z sin A, y = a sin A z cos A, 



and c, x; d, w corresponding functions of angles B, C. It remains only to 

 satisfy y 2 +x 2 +w 2 = a 2 , viz., 



a 2 (S sin 2 A - 1) -azS sin 2A+Z-2 cos 2 A = 0. 

 The discriminant must be a square, whence 



& 2 = 2S cos2A+2S cos (A-B). 



Take C = A+B- 90. Then 2 = sin 2A sin 2B. Take sin 2A = tan B/2. 

 Then fc = sin 4A/(l+sin 2 A). The case cotA/2 = 2 leads to the solution 

 [due to Euler, 32 58]: 



2 = 186120, y = 23838, x = 102120, w = 32571. 

 Bills gave also 280, 105, 60, 168 and 1120, 3465, 1980, 672. 



Judge Scott 38 found 639604, 3456000, 3750000, 832797 [due to Euler, 32 

 55]. 



S. Tebay 39 gave the solution x 2 , , u 2 , where 



z = (s 2 -l)(s 2 -9)(s 2 +3), 2/ = 4s(s-l)(s+3)(s 2 +3), 

 z = 4s(s+l)(s-3)(s 2 +3), w = 2s(s 2 -l)(s 2 -9). 

 A. Martin 390 gave a complete solution by the method of Tebay. 39 



THREE SQUARES WHOSE DIFFERENCES ARE SQUARES. 



Under Euler 28 of Ch. XV are cited various papers on the related problem 

 to make xy, xz, yz all squares. 



L. Euler 40 made the differences of x 2 , if, z 2 squares by taking 



x_p 2 +l y 



_ ^ 



z~p 2 -!' z~q 2 -!' 

 whence x 2 z 2 and y 2 z 2 are squares. Also x 2 y 2 = D if 



Each factor will be a square if 



_a 2 +b 2 q_c z +d 2 

 Pq= ~2ab~' p~ 2cd 



The product of the latter must be a square q 2 . Take a, b =/d=gr; c,d = hk. 

 Then must (/ 4 -^ 4 )(/i 4 -A; 4 ) = D [cf. Euler 28 of Ch. XV.] 

 J. Cunliffe 400 treated the problem. 

 "Calculator" 41 took 

 x = (r 2 + s 2 ) (m 2 + n 2 } , y= (r 2 + s 2 ) (m 2 -n 2 ), z = 4rsmn - (r 2 - s 2 ) (m 2 - n 2 ) . 



Then x 2 y 2 and x 2 z 2 are the squares of 2mn(r 2 +s 2 ) and 



2rs (m 2 n 2 ) + 2mn (r 2 s 2 ) . 



38 Math. Quest. Educ. Times, 16, 1872, p. 108. 



39 Ibid., 68, 1898, 103-4. 

 390 Ibid., 24, 1913, 81-2. 



40 Algebra, 2, 1770, 236-7; 2, 1774, pp. 320-7; Opera Omnia, (1), I, 473-7. 

 400 The Math. Repository (ed., Leybourn), London, 3, 1804, 5-10. 



41 The Gentleman's Math. Companion, London, 3, No. 14, 1811, 334-6. 



