CHAP, xxii] EQUAL SUMS OF BIQUADRATES. 653 



quadrate expressed as a sum of r+5 biquadrates. In fact, 



where r = c 8 6 8 , 2 = xl+x*+- +x*. 



EQUAL SUMS OF BIQUADRATES. 



A. Martin 200 tabulated various sets of numbers having equal sums of 

 fourth powers, as 1, 2, 9 and 3, 7, 8; 1, 9, 10 and 5, 6, 11 ; 1, 11, 12 and 4, 9, 

 13; 1, 5, 8, 10 and 3, 11. 



C. B. Haldeman 201 noted that the sums Q a , b and Qd, e of three biquadrates 

 are equal if 3a 2 +6 2 = 3d 2 +e 2 . Taking e = b v, we get b, e rationally in 

 terms of a, d, v and see that 



is unaltered by the interchange of a and d. For a = l, d = v = 2, we get 



8 4 +9 4 +17 4 = 3 4 +13 4 +16 4 . 

 Next, let 



2s 2s ' 



whence 6 2 = N 2 /(4s 2 ), N 2 = d 4 -s 4 -12aV. Taking 



N = d 2 -2p 2 s 2 /(3g 2 ), d = v+3aqfp, 



we get a and d rationally. Or take N = d 2 s z , whence d 2 = 6a 2 +s 2 , set 

 a = 2s+y and solve as usual. Again, 



if 



Take a = l+x, d = %+y and solve as usual. Finally, to find a sum of four 

 biquadrates equal to a sum of three, employ his 239 identity (1) and equate 

 the left member to Q m , . The resulting condition, 3(3a 2 +Z 2 ) 2 = 

 is satisfied if 



ra = 



A. Cunningham 202 noted that Z 4 + F 4 +^ 4 +2 4 = Zj+4 +y\+z\ follows by 

 combining a solution of each of Z 4 +F 4 =Zt+Ft, x*+Yi+z*=x* +y\+z\. 

 Again, x*+y 4 +2u 4 l =xl+y* l +2u 4 follows from 



(solved, Cunningham 240 ) with u = A 2 +3B 2 ,u 1 = A 2 l +3B 2 l ,AB = A l B 1 , whence 

 zz\. 



A. S. Werebrusow 203 gave an incorrect proof of the impossibility of 

 = 3w 4 in relatively prime integers. 



200 Math. Magazine, 2, 1896, 183. 



201 Ibid., 2, 1904, 286-8. For the notation Q, see Haldeman 196 (4). 



202 Messenger Math., 38, 1908-9, 103-4. 



203 L'interme'diaire dee math., 15, 1908, 281. Cf. 16, 1909, 55, 208; 17, 1910, 279. 



