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



To make x 2 +y 2 +2z 2 , x 2 +z 2 +2y 2 , y 2 +z 2 +2x 2 all squares, A. M. 

 Legendre 58 set y = x+2p, z = x+2q. Then x~+y 9 ~+2z 2 = 4:(x+f) 2 for 

 (2fp 2q)x = p 2 -{-2q 2 f 2 . Equating this to the value found similarly 

 from x 2 +z 2J r2y 2 = D, he was led to the values 



x = 7p 2 -30pq+7q 2 , y = 23p 2 -Upq+7q 2 , z = 7p 2 -Upq+23q 2 . 

 Substitute these into y 2 +z 2 +2x 2 and set p/q = 1 + 6. Then shall 



IfiQ 



. 



The particular solution = 208 gives a; = 18719, y = 62609, 2 = 18929. 



T. Pepin 59 noted that also 0= 1 and 2 (whence x = y = 7, 2 = 23; 

 x = y = 2 = 1) and applied his first formulas (Ch. XXII 157 ) with i = 0, 

 z 2 = 1, 3= 2 and found 0= 8/15, whence x : y : z = 77 : 77 : 253. 



C. Gill and W. Wright 60 made x 2 +y 2 +z 2 +v 2 , x 2 +y 2 -z 2 +v 2 , x*-y* 

 -}-z 2 -{-v 2 , y 2 +z 2 x 2 +v 2 squares. To satisfy the second and third conditions, 

 take 2vx = y 2 z 2 , say 2v = y-\-z, x = yz. The fourth condition holds if 

 y 2 +Wyz- J rz 2 = D = (y p) 2 , which gives y. Clearing of denominators, we 

 now have 



y = 2p 2 -2z 2 , v = $z 2 +2pz+p 2 , x = 2p 2 -4pz-22z 2 . 



Then the first condition leads to a quartic in p; equating it to (3p 2 2pz-\-d)-, 

 we get d=-23z 2 /3. 



To find four squares the double of whose sum is a square, and double 

 the difference between the sum of any three and the fourth is a square, 

 they took (x-\-y) 2 , (x y} 2 , v 2 , z 2 . Then two conditions are satisfied if 

 v+z = 4:X, vz = y, and the solution follows readily. 



The solutions of the system 2x 2 +2y 2 3z 2 = D, etc., and the system 

 x 2 -\-2(y 2 z 2 ) = D, etc., offer no special interest. 



To find three numbers such that the square of each plus the product 

 of the same number and the sum or difference of the remaining two gives a 

 square, several 61 used the numbers a 2 , b 2 , c 2 . Then the conditions reduce 

 to a 2 +6 2 +c 2 = D, a 2 +c 2 -6 2 = D, a 2 +6 2 -c 2 = D. To satisfy the first two, 

 take b 2 = 2ac. Equate the third to (en a) 2 . Take n= 3/4. 



A. Gerardin 62 treated the system N = Ph 2 -k 2 (P = n+l, ?i+2, , 

 tt+a). 



E. Fauquembergue 63 made the four functions x 2 hy 2 , u 2 hy 2 squares. 



H. Holden 63a showed that 



68 Theorie des nombres, 1798, 460-1; ed. 2, 1808, 433^; ed. 3, 1830, II, 125; German transl. 

 by Maser, II, 122. J. Cunliffe, New Series of Math. Repository (ed., T. Leybourn), 

 1, 180(3, I, 189-191, used the same method with 2p 2q, 2p replaced by m, n, and 

 obtained an equivalent result. 



Atti Accad. Pont. Nuovi Lincei, 30, 187G-7, 219-20. 



60 The Gentleman's Math. Companion, London, 5, No. 30, 1827, 579-83. 



61 Ladies' Diary, 1833, 38-39, Quest. 1547. 



L'interm<5diaire des math., 23, 1916, 88-93. He gave 139 examples. 



83 Ibid., 24, 1917, 38-9. 



630 Messenger of Math., 48, 1918, 77-87, 166-179. 



