652 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxn 



E. Barbette 198 used the final method of Martin 194 to show that (2) is 

 the only sum of distinct bi quadrates ^=14 4 equal to a biquadrate, and that 



4 5 +5 5 +6 5 +7 5 +9 5 +ll 5 = 12 5 



is the only sum of distinct fifth powers ^ll 5 equal to a fifth power. 

 R. Norrie 199 found (in confirmation of Euler's 190 conjecture) 



(5) 353 4 = 30 4 +120 4 +272 4 +315 4 , 



by a series of special assumptions which lead to this single result. Next 



(p. 77), 



provided [see (4)] 



To solve the latter, set u = rxi+2, v=l, x = rx z -\-3, y = rx z +l. Hence 



Equate the coefficient of r to zero. Then the equation gives r. For 6 

 biquadrates (p. 80), use 



- F 8 ), 



From the latter, 



Z2 r+s_ y 2r+ 3 = 2 (2z z/) 4 (z 8 + 1 6?/ 8 ) (X*+ F 4 ) (X & + Y 8 } - " (X* r+2 + F 2r+2 ), 



the second member being double the sum of 2 r+2 biquadrates. Hence 

 (X 2r+2 +Y 2 " +2 y equals a sum of 2 r+2 +2 biquadrates. Returning to the 6 

 biquadrate case, take x = u?, y = 2iP, whence x 8 +16y 8 equals the value of 



, , , {& 3 (6 4 -3c 4 ) } 4 + (c 3 (c 4 -3fr 4 ) } 4 + (2bc(6 4 -c 4 ) } 4 (b 4 +c 4 ) 



(6 4 +c 4 ) 4 



for 6 = it 2 , c = 2y 2 . Thus we get a biquadrate expressible simultaneously 

 as a sum of 6, 8 or 10 biquadrates. The sum of two of these biquadrates 

 has the factor u s +16^ 8 , which as before can be replaced by the sum of four 

 rational biquadrates. In this way we can assign a biquadrate which is a 

 sum of any even number >4 of biquadrates. 



For 7 biquadrates (p. 84), take t=(x*+y*+z 4 )/8 in 



and set x = p 2 q*, y = 2pq+q 2 , z = 2pq-\-p z . We get a relation between 

 biquadrates, one with the coefficient 2, for which we substitute the sum of 

 three rational biquadrates given by Gerardin's 208 (3). Again, 



where T=(x-\-y)*-\-(x ?/) 4 . But for any r>2, we can express T (in one 

 of its two occurrences) as a sum of r biquadrates and hence obtain a bi- 



198 Les sommes de p-i&nes puissances distinctes 6gales It une p-ieme puissance, Liege, 1910, 



133-146. 



199 University of St. Andrews 500th Anniversary Memorial Vol., Edinburgh, 1911, 89. 



