666 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



L. Euler 294 required four positive integers whose sum and sum of squares 

 are biquadrates. He took them to be z = a 2 +fr 2 +c 2 d 2 , y = 2ad, z = 2bd, 

 v = 2cd. Then Sz 2 =(Za 2 ) 2 . Set a = p 2 +g 2 +r 2 -s 2 , b = 2ps, c = 2qs, d = 2rs. 

 Then Za 2 = ( Sp 2 ) 2 . It remains to make Zz = D 2 . Take p = s - q + fr . Then 



Vzz = 2q*-3qr-2qs+r*+5rs+2s 2 , 

 which for q = r+t will be the square of 3r/2 u if 



For g = r = 2, s = 9, p = 10, we get z = 409, ?/ = 24, 2 = 160, y = 32, 



2x 2 = 21 4 . Euler gave a similar treatment of the problem in five integers. 



Euler 55 (first paper of 1780) treated the problem for n = 3, 4, 5 and 

 obtained the sets 8, 49, 64; 320, 400, 961; 16, 48, 104, 193; 32, 32, 88, 137; 

 16, 16, 32, 72, 89; 64, 152, 409; 17424, 108864, 580993, the last two sets 

 having also the sum a biquadrate. 



J. Cunliffe 294 " took z 2 , 2xy, 2y 2 as the n = 3 numbers, the sum of their 

 squares being (x 2 +2?/ 2 ) 2 . Their sum is the square of ryx if y = 2r-{-2, 

 x=r z -2. For r = v-3, z 2 +2?/ 2 =(i; 2 -6y-9) 2 if y = 28/5. * 



_ 



Walmond and Mason 295 wrote x* for the biquadrate. Take r= V4z 5, 

 2x 1 and x 2 2 as the n = 3 numbers, their sum being (x+1) 2 if r 3 = 1, 

 whence z = 21/4. For n = 4, take r= V6z 6, x2, x 1, x z 1, whose 



sum =(z+l) 2 if r-4 = l, z = 31/6. Forn = 5, taker= V4x-12, x+1, x-l, 

 2x-l, x 2 -3, whose sum =(a;+2) 2 if r-4 = 4, x = 19. 

 S. Bills 296 employed the identity of Aida 59 of Ch. IX: 



, 



Vi-i (i = 2, - -, n), 



The remaining condition u\-\ -+u n = D becomes a quartic in x n which 

 is equated to the square of xl+2x n -iX n +xl-\ ----- h^Li- Hence 



3 



X n =T ~2X n i } 



where r = XiH +x n _ 2 . 



A. B. Evans 297 used the numbers x, aiy, , a n -\y and wrote 



m = a z -\ ----- \-a n -i, v = a\-\ ----- |-oLi. 



Take x = a? py. Then x 2 +(a 2 } +v)y'* = a* determines y rationally. Hence 

 a~ 2 (ai+y+p 2 ) 2 Dc+(ai+w)2/]=(a?+pai+&) 2 , b = pm-\-v |p 2 , determines ai 

 rationally. 



D. S. Hart 298 used the numbers px z -ax, px*+ax, -, Nx-Zx, Nx 2 +Zx 

 and, if n is odd, Sx z . Equating the sum of their squares to (xm/ri)*, we get x 2 . 



294 Opera postuma, 1, 1862, 255 (about 1782). 



2940 N GW Series of Math. Repository (ed., T. Leybourn, 3, 1814, I, 79-80. 



296 Ladies' Diary, 1827, 36-7, Quest. 1452. Reference was made to *Ferussac, Bull, des Sc. 



Math., Ill, 276. 



896 Math. Quest. Educ. Times, 18, 1873, 104-5. 

 *" Ibid., 22, 1875,69-71. 

 8 Ibid., 24, 1876, 55-57. 



