CHAP. XXII] SUM OF BIQUADRATES A BlQUADRATE. 651 



To find a sum of n biquadrates equal to a biquadrate for n = 9, 10, 11, 12, 

 multiply (3) by a suitable biquadrate and eliminate one biquadrate by 

 use of one of the earlier results. Finally, given 



2 4 -t-6 4 +8 4 +2 4 +7 4 +12 4 = 13 4 , 2+6 = 8, 



we can find a, b so that 2 4 +6 4 +8 4 = a 4 +6 4 +(a+b) 4 = 2(q 2 +ab+b 2 ) 2 . Thus 



= 52, o = i(-&- V208-362). Set b = 



208-3& 2 =-37/ 2 -36?/+100 = | 



\ i / 



whence we get y, b, a. Take s = 2, t = l. Then 7y= 76, 76= 34, 

 7a = 58 and 



() 4 + () 4 + () 4 +2 4 4-7 4 + 12 4 = 13 4 . 



A. Martki 197 employed methods admittedly similar to Haldeman's, 

 whose manuscript was in his hands, but found many new sets of biquadrates 

 whose sum is a biquadrate. For 5 biquadrates, take 



which reduces to 2e(3a 2 +6 2 ) = y(y 2 e 2 ). First, take y = 2e; then 



if = 



which for e = 2(s 2 +3 2 ) leads to Haldeman's (3). For y = 3e, we get a 

 result equivalent to the last. The next solvable case is y = 8e, giving 



+ (24s 2 -96s-72* 2 ) 4 +(63s 2 +189 2 ) 4 =(65s 2 +195Z 2 ) 4 . 

 For y 13e, we get a similar formula. Next, let 



The first becomes w 2 +z 2 = s 2 , whence take z = 2pq, w = p 2 q 2 , s = p 2 -{-q 2 . 

 The case p = 2, q = l, leads to (3). Omitting the discussions found to be 

 unfruitful, let p = r + 2q, x = t-\-2q 2 . Then 



y* = wz-3x 2 = 2qr 3 +12q 2 r 2 -3t z +A, A = 22q 3 r-12qH. 



Take A = 0, whence t= llqr/Q. Set y = qrmfn. We get q in terms of ra, n, 

 whence 



+ (/5 2 -576n 4 ) 4 = (j3 2 +576n 4 ) 4 , = 12m 2 -23n 2 , /3 = 12m 2 +25n 2 . 



In the Congress paper, on the contrary, he took t = 4q and found the special 

 solution 2 4 +13 4 +32 4 +34 4 +84 4 = 85 4 . For n biquadrates, n = 6, 7, 8, 11, 

 he took 



13 4 , Q a>6 +2 4 +4 4 +5 4 +8 4 = 9 4 , 



= 9 4 , Qa,H-Qc,H-Q e ,H-7 4 +14 4 = 21 4 , 

 and found other sets by combination. 



197 Deux. Congr&s Internal. Math., 1900, Paris, 1902, 239-248. Reproduced with additions 

 in Math. Mag., 2, 1910, 324-352. 



