658 HISTORY OP THE THEORY OF NUMBERS. [CHAP, xxn 



From a given solution is deduced a second by long formulas, whence 

 l 4 +3 4 +10 4 = 2-71 2 , 7 4 4-7 4 +12 4 = 2-113 2 , ! 4 +l 4 +2 4 = 2.3 2 . 



A. Martin 231 gave 9 biquadrates, 720 4 , -, 3120 4 , whose sum is a square. 



Martin 232 , assuming that the sum of the fourth powers of x, x ay, 

 x by, x cy, is a square, obtained x[y = a/!3, where a and /3 are polynomials 

 in a, b, c, and took x = a, y = ft. By the same method, he 233 elsewhere found 

 199 4 +271 4 +343 4 +559 4 = 344162 2 . 



Martin and R. J. Adcock 234 repeated the solution by Diophantus and 

 stated that Diophantus' result 12 4 +15 4 +20 4 = 481 2 gives the least solution 

 in integers. 



E. Fauquembergue 235 noted that, if a 2 +/3 2 = 7 2 , 



M) 4 +(07) 4 +(7) 4 = (c* 4 +a 2 /3 2 +/3 4 ) 2 , 

 (2a 2 jS7 3 ) 4 +(2a/3V) 4 +[(^ 



These two formulas were given also by A. Martin. 236 To find n biquadrates 

 whose sum is a square, the latter took their roots to be x, xay, x by, 

 piy, "-, p n -*y. Then shall 



say the square of 2x z Za xy + \ { 6 Sa 2 ( Sa) 2 } y 2 . Thus x/y is determined . 

 E. B. Escott 237 noted that 



(m 2 + mn + n 2 ) 4 (mri) 4 (mn + n 2 ) 4 = [m (m + n) (m 2 + ?nn + 2n 2 ) ] 2 . 

 E. Fauquembergue 238 gave identities including 



(a 4 +26 4 ) 4 = (a 4 -26 4 ) 4 +(2a 3 6) 4 +(8a 2 6 6 ) 2 



= (2a 2 6 2 ) 4 +(2a 3 6) 4 +(a 8 -4a 4 6 4 -46 8 ) 2 . 

 C. B. Haldeman 239 found four biquadrates whose sum is a square: 



Take s = d?-\-v, 3a?-\-b 2 = vg. Then v, b 2 , s are determined rationally in 

 terms of d, g, a. Take g = 2, a = 3/7. Then 6 2 = 4d 2 /7- 27/49. Since b is 

 rational for d=l, take d = y-\- 1 and equate b to ry/t+1/7, thus determining y. 

 Then 



6= -(7r 2 -56rf+4* 2 )/(7/b), d= (7r*-2rt+4t 2 )/k, k = 7r*-W. 



For r= 1, = 0, we get 2 4 +4 4 +6 4 +7 4 = 63 2 . Next, let the sum of the initial 

 biquadrates equal 2s 2 . The condition is evidently satisfied if 



s= 



231 Annals of Math., 5, 1889-90, 112-3. 



232 Ibid., 6, 1891-2, 73. 



233 Amer. Math. Monthly, 1, 1894, 401-2. 



234 76^., 279-80. 



238 L'interm6diaire des math., 1, 1894, 167 [6, 1899, 186]. 



238 Math. Magazine, 2, 1898, 210-1. 



237 L'intermddiaire des math., 6, 1899, 51. 



238 Ibid., 7, 1900, 412. 



239 Math. Magazine, 2, 1904, 285-6. 



