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



(iv) The first two equations (1) are satisfied if z = mn(A B),y+x = 2m 2 A, 

 yx = 2n-B, p = mn(A+B) ) q = mn(a 2 2ab b 2 ), where A=a(a+6), and 

 B = b(a b). The third equation (1) becomes 



Set m=f+g, n=f-g, r=(A+B)f 2 +4(A-B)fg-(A+B}g 2 . Then 



/ :g = B--A 2 :2AB. 



A. M. Legendre 47 noted that the last two conditions (1) are evidently 

 satisfied if 



z = r 2 +s 2 , y = r 2 -\-rs s 2 , z = r 2 rs s 2 . 



Then the first condition becomes r 4 4r 2 s 2 +s 4 = D. Set r = s(2+0) and 

 make the quartic function of < the square of l+80+o;0 2 . The case a = l 

 gives <f>= 23/4, r = 15, s = 4, whence 2 = 241, ?/ = 269, 2 = 149, which is 

 apparently the least solution. 



J. Cunliffe 48 noted that (1) give x 2 = ^(g 2 +r 2 ), etc., whence 



Hence, if we set x = y+pv, q = p-\-av, we get v=(2<rp 4p?/)/(2p 2 o- 2 ). To 



satisfy 2y 2 p 2 = r 2 , set 



y = D(m 2 +n 2 ), p = D(n 2 -m 2 +2mn), r = D(m 2 -n 2 +2mri), D = 2 P 2 -<r 2 . 



The resulting value of |(p 2 +? 2 ) will equal the square of 



where ^L =2 P 2 +2 P (r+^ 2 , if 

 m : n = 



Taking p = o- = l, he obtained, as his least answer, = 149, ?/ = 269, 2 = 241. 

 D. S. Hart 49 noted that (1) are equivalent to 2r 2 +2q 2 = D, 2r 2 +2p 2 = D, 

 2g 2 +2p 2 =D. The first is satisfied if r = p 2 -2cr 2 , g = p 2 +4pcr+2(T 2 . Set 

 p = l-\-r, a = p 2 +2po-+2<j 2 . Then the last two conditions of the problem 

 become 2Z 2 +4rZ+4r 2 = D, 2Z 2 +4rZ+4a 2 = D. Equating the latter to 

 (2a lty, we get I in terms of t, r, a. Then the former becomes a quartic 

 in t. S. Bills satisfied the first two of Hart's conditions by taking 



^ 

 r ' P '~R 2 -S 2 -2RS'' 



The third condition leads to a quartic. 



G. B. M. Zerr 50 took x-z 1 , ifz 2 and z 2 as the squares and set 



(A) x 2 +y 2 -! = (t+u} 2 , x 2 -y 2 +l = (t-u) 2 . 



Since x 2 = t 2 +u 2 , take t = n(p 2 q 2 }, u = 2npq, whence x = n(p 2 -\-q 2 }. Take 



7 Th6oric des nombres, 1798, 461-2; ed. 2, 1808, 434; ed. 3, 1830, II, 127; German transl. 



by Mascr, 2, 1893, 124. 

 The Gentleman's Diary, London, No. 62, 1802, 41-2, Quest. 823. Math. Repository (ed., 



Leybourn), 3, 1804, 97. 

 " Math. Quest. Educ. Times, 20, 1874, 84-6. 

 Amer. Math. Monthly, 10, 1903, 207-8. Cf. papers 114-5 of Ch. XVI. 



