CHAP. XXII] BlQUADRATES AND SQUARES. 657 



where 



p = s 3 +9s*t+lSst 2 , s 3 +12s 2 +36s* 2 +36 3 = P 2 +Q 2 , (P 2 +Q 2 )v 2 = E7 2 +7 2 . 



RELATIONS INVOLVING BOTH BIQUADRATES AND SQUARES. 



Diophantus, V, 32, treated x*+y*+z* = v 2 by setting v=x 2 k. Then 

 x z = (k z -y*-z*)((2k). Take k = y 2 +z*. Then X 2 = y 2 z 2 /(y 2 +z 2 ). Hence 

 2/ 2 +2 2 equals a square w 2 . For 7/ = 3, 2 = 4, we get k = 25, x = 12/5. Dio- 

 phantus' method thus leads to the identity (cf. Fauquembergue 235 ) 



Taking y ab, z = bc, w = ac, we get [Norrie, 211 p. 91] 



a 4 +6 4 +c 4 == (a 2 -6 2 +c 2 ) 2 , a 2 6 2 +6 2 c 2 = aV. 



E. Waring 226 reproduced Diophantus' argument with k eliminated. 



F. Proth 227 recalled that any prime N of the form Qx +1 is expressible in 

 the form N=a?+b 2 +ab. Thus 2N = a?+b 2 +(a+b) 2 . By multiplication, 



= a 4 +6 4 +(a+6) 4 , whence 



It is stated that if N is of the form 6z+l, whether prime or not, 2N 2 is a 

 sum of three biquadrates [incorrect, Kempner 42 of Ch.XXV, Diss., p. 44]. 

 If N is expressible in two ways in the form a 2 -f 6 2 +a6, as 



we get a number expressible as a sum of three biquadrates in two ways : 



2-91 2 = 5 4 +6 4 +H 4 =l 4 +9 4 +10 4 . 



S. Realis 228 noted that z\+z\+z\ = 3z* if 



2! = 5s+2a/3(2 2 +5/3 2 ) +9a 2 /3 2 , Z 2 = 5s+2a|3(5a 2 +2/3 2 ) +9o; 2 j 3 2 , 

 z 3 = 5s+16a j 3(a 2 +/3 2 )+27a 2 /3 2 , z =*{25 3 +72a 2 /3 2 (a+j3) 2 }, 



where s = a 4 +/S 4 , ^ = a 2 +aj34-/3 2 . 

 G. Dostor 229 gave the identity 



(a+b+c-dy+(a+b-c+dy+(a-b+c+dy+(-a+b+c+d)* 



S. Realis 230 noted that v 4 +z 4 +2/ 4 = 22 2 is satisfied if 



x = 2057a 3 - 2541a 2 /3+2787a/3 2 - 391/3 3 , 

 y = 391a 3 - 2787o: 2 /3+2541a 1 3 2 - 2057/3 3 , 

 v= (2a+2 j 8)(391a 2 -730a/3+391^ 2 ), 

 whence for a = l, /3 = or 1, 



46 4 + 121 4 +23 4 = 2 10467 2 , 26 4 +239 4 +239 4 = 2 57123 2 . 



224 Meditationes Algebraicae, 1770, 194; ed. 3, 1782, 325. 

 127 Nouv. Corresp. Math., 4, 1878, 179-181. 

 /&., 350. 



229 Archiv Math. Phys., 60, 1877, 445. 



230 Nouv. Corresp. Math., 6, 1880, 238-9. Misquoted, C. A. Laisant, Algebre, 1895, 221-2. 

 43 



