INDETERMINATE QUARTIC EQUATIONS 65 



If then we multiply corresponding sides of these last two 

 equations together it is clear we shall obtain an identity of 

 the form 



The foregoing processes obviously admit of infinite com- 

 bination and repetition. 



Third method. We have identically 



and 



Hence we shall have 



provided z 2 +3?/ 2 =w 2 +3v 2 (2) 



Now the most general solution of (2) is given by 



where x is 



Hence the most general solution of (1) is given by 



-H4XI/) 4 



Thus X= 2/=2, v=---l gives 



8 4 + 11 4 + 19 4 =1 4 + 16 4 + 17 4 . 

 Cor. 1. From the foregoing we may derive the solution of 



= (x n + y n ) + (x n y n 

 For it is clear that the integer N, which is equal to the 

 product 



(a 1 2 +36 1 2 )(a 2 2 +36 2 2 ) . . . (a r 2 +36 r 2 ), 



\ A x / \ A ft / \ / ^ 



is expressible in the form > 2 +3<Z 2 in 2 r " 1 ways, and therefore 

 as above 2N 2 is expressible in the form (x+y)*+(x 2/) 4 +(2i/) 4 

 in the same number of ways. 



In practice, where an arithmetical result merely is desired, 

 it is easier to proceed as follows. Selecting the smallest 



i 



