INDETERMINATE QUARTIC EQUATIONS 79 



we may derive others from it of the same form by multiplying 

 by (p 2 +3# 2 ) 4 . For we then have 



\Po(P 2 + 3<z 2 )!H^i(P 2 + 3<Z 2 )! 4 +!P 2 (P 2 + 3<? 2 )S 4 + 2(* 2 + 3</ 2 ) 2 



(FM-3<? 2 ) 4 (9) 



But 2(ic 2 +32/ 2 ) 2 (p 2 +3^ 2 ) 4 is expressible in the form 

 2(^1 2 +3J5 2 ) 2 , where 



A=(p*-3q 2 )x+6pqy, B=2pqx-(p*-3q*)y, 

 (A and B having in general a variety of values depending 

 on the composite character of x z +3y 2 and of 

 Hence (9) becomes 



and since p and q are arbitrary this will give an infinity of 

 solutions. 



In practice, where an arithmetical result merely is desired, 

 we proceed as follows. Starting from any solution of the 

 required form, say 



5 4 =4 4 +3 4 +4 4 +2 4 +2 4 



=4 4 +3 4 +2-12 2 . 



we multiply this by 7 4 say, 7 being the smallest integer of the 

 form > 2 -f 3<? 2 , and obtain 



35 4 =28 4 + 21 4 + 2- 12 2 (1 2 + 3-4 2 ) 2 

 =28 4 +21 4 +2(24 2 +3-2 2 ) 2 

 =28 4 + 21 4 + 26 4 + 22 4 +4 4 . 

 A second application of this process gives 



245 4 = 196 4 + 188 4 + 147 4 + 142 4 +46 4 . 



Similar results may be obtained by multiplying by 13 4 , 

 19 4 , etc. 



The defect of all the foregoing solutions lies manifestly in 

 the assumption that P 3 -fP 4 =P 5 , a restriction which such 

 identities as 



31 4 =30 4 + 17 4 + 10 4 + 10 4 + 10 4 , 

 313 4 =312 4 + 90 4 + 75 4 + 70 4 + 30 4 , 

 etc., show to be unnecessary. 



