INDETERMINATE QUARTIC EQUATIONS 91 

 Hence we may take 



u=x 2 +y 2 , v=2xy 

 and with these values equation (4) becomes 



Thus x=2, y=l gives 



481 2 =12 4 +15 4 +20 4 . 



Cor. The foregoing result evidently amounts to this, that 

 if a 2 b 2 +b 2 c 2 =c 2 a 2 , then 



Again, to solve (3) when X =2, n=3, x^^ ^3 1 since 



a 4 + 6 4 + (o+ &) 4 = 2(a 2 + a&+ 6 2 ) 2 , 

 replacing a and &, each by its square, we have 



(a 2 + 6 2 ) 4 = J[a 



If then we put a=x 2 y 2 , b=2xy, this becomes 



(x 2 +y 2 )*+ (2xy) 8 + (x 2 -y 2 )* 



4 ] (6) 

 by (5) above. Thus x=2, y=l gives 



3 8 +4 8 +5 8 =2(12 4 +15 4 +20 4 )=2-481 2 . 

 The last part of (6), which is simply a solution of 



is immediately obtained otherwise. For if a, 6, c be integers 

 connected by the equation a 2 +6 2 =c 2 , we have, on squaring 

 each side of this equation 



and on again squaring we obtain 



a 8 + 6 8 + c 8 = 2(a 4 6 4 + 6 4 c 4 + c 4 a 4 ) 



which is the required result if we put a=x 2 y 2 , b=2xy, 

 c=x 2 +y 2 to make 2 +6 2 =c 2 . Corresponding results may 

 naturally be obtained by squaring any identity of the form 



But for all values of n greater than 4, a single algebraical 



