CHAP. XXI] TWO EQUAL SUMS OF TWO CUBES. 553 



there exists the simpler solution x = 6, 2/ = 5. To treat (4), he set 



(5) A=p+q, B = p-q, C = r-s, D = r+s. 

 Thus 



(6) p(p 2 +3(? 2 )=5(s 2 +3r 2 ). 

 Taking 



p = ax-+-3by, q = bx ay, s 

 we have 



Hence our equation becomes p(ax+3by} =y(3cy dx), whence 



x= 3n6j8+3nc7, y 

 Writing X, ju = Sac 360^ ad +36d, we get 



The abbreviatons 0, 7, X, n were not used by Euler ; but their introduction 49 

 enables us to point out the identity which underlies his solution. In 



it is the final factor which vanishes, and this in view of the identit 



which in turn follows from 



Euler noted (p. 206) that we may solve similarly Z:r = Xp, where 

 Tr = mp 2 -{-nq 2 , p = mr 2 -\-ns 2 , while I, X are any linear functions of p, q, r, s, 

 by setting 



p = nfx+gy, q = mfy gx, r 

 Then 



Hence x[y is ratonal. 



Euler 50 treated (4) by setting, without loss of generality, 



A = (m ri)p+q 2 , B=(m+ri)p q 2 , 

 C = p z (m+ri)q, D = p 2 +(m-ri)q. 



Then (A+B)(A 2 -AB+B 2 ) = (D-C)(D 2 +DC+C 2 } becomes, after division 

 by 2m(p 3 -g 3 ), ra 2 +3n 2 = 3pg. Thus m = 3k, where pq = n?+3k 2 . But he 

 had proved in the same paper that every divisor of n 2 +3/c 2 , in which n 

 and k are relatively prime, is of like form. Thus 



while n is ac^FSbd or its negative. 



"L. E. Dickson, Amer. Math. Monthly, 18, 1911, 110-111. 



60 NoviComm. Acad. Petrop., 8, annees 1760-1, 1763, 105; Comm. Arith., I, 287; Opera 

 Omnia, (1), II, 556. 



