73 

 The general solution of the equation 



,2 



may be obtained by a similar method. The two factors x + y 

 and (cc + yP — Zxij of or^-\-y^ can, if x and y are prime to one 

 another, have no common factor but 3 : hence we have either 



x + y = m^, 

 or x + y = ?>m^. 



In the first case we put 



i.e., x = 2}^ — q'^, y^^pq — q"^. 



We thus have 



the solution of which is 



'p = ^r^ — 2rs + s^j q = 2rs, m = 3r^ — s^. 



This type of solution of 



x^ + y3 = ^2 



is therefore 



x = fr-sj (3/- - sj (3r2 + s^) , 



y = 4:rs(3r^-Srs + s^), 



z=\3r^-s^\ (9r4 - ISr^s + ISrH^ - 6rs^ + s*) . 



Particular cases are 



83-7^ = 132, 56^ + 653 = 6712, 



105^ - 104^ = 18P, 573 + 112^ =: 12612. 



Taking now the second case, in which 



x + y = Sm'^, 

 we may put 



x + (x)y = (2+ wj fp + (oqj^ ; 

 whence 



x = 2p^ — 2pq — q^j y = p^ + 2pq-2q^, 

 and therefore ni^ = p^ — q^^ 



so that p = r'^ + s'^y q = r^ — s^, m = 2rs, 



and the second type of solution of 



x^ + y^ = z^ 

 is x= -r^ + 6r2s2 + 3.5*^ 



y = r4 + 6r2s2-3s*, 



Particular cases are 



V + 2^ = 3\ lP + 373=:2282, 7P-233 = 32. 14*. 

 Tlie solution of 



x'^ + y'^ = 2z^ 

 is required below. The only case which leads to a solution is 

 x + y = Qm.^, 



x+ (oy = {2+ u)) fp-\- cog/, 

 i.e., x = 2p^ -2pq-q^, y = p^ + 2pq-2q^, 



p2_g2-_2m». 



