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 



PS+PJ+ . . . +P 7 4 =PY+P' 2 4 + . . . +P' 7 4 . 



The foregoing processes obviously admit of infinite com- 

 bination and repetition. 



Third method. We have identically 



and (u+v)*+(u-v)*+(2v)*=2(u 2 +3v 2 ) 2 . 



Hence we shall have 



(x+y)*+ (x-y)*+ (2y)*=(u+v)*+ (u-v)*+ (2w) (1) 



provided x 2 +3y 2 =u 2 +3v 2 (2) 



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



h arbitrarv 



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

 [(3x a +l)v+(3x 2 +2x-l)i/] 4 +[(3x 2 +l)t;+(3x a -2x-l)i/]* 



Thus X= y=2, v=l gives 



8 4 +ll 4 +19 4 =l 4 +16 4 +17 4 . 

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



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 ), 



is expressible in the form p 2 +3q 2 in 2 r ~ 1 ways, and therefore 

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

 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 



r 



