74 

 Whenoei p = r^-\-2s^^ q = r^ — 2s^, m = 2rs; 



y = r4+12r2s2_i2s4,' 



e.g., 23^ + P = 2.78^ 239^-143^ = 2.23162. 



(A form of solution of x^ + y^ = az^ is obvious: we write 

 ^as^, ?>a^s^ instead of 125^, 12s* in the above result.) 



The last result enables us to solve the equation 



x^ + y^ = 2'^. 



If X, y, z are prime to one another, z^ — x and z^ + x have at 

 most a common factor 2. If they are both odd we put 



^ = i fP^ — ^^)y y^PQ, p^-{-q^ = 2z^. 



Substituting for p, q, and z the values which satisfy the last 

 equation, we obtain 



x^lUr'^-s^l (144^8 + 408risi + 5-8), 

 y = (12r4 + 12r252 _s<^) ( _ 12r4 + 12r2s2 + s*), 

 z = 6rs (12r^ + s^) ; 

 e.g., 60832 + 233 = 78S 4332+143^ = 42^ 



Tlie case in which z^ + x and z^ — x are both even appears 

 not to lead to a solution. 



x^ + y^ 



The equation 



also admits of solution. 



We put x + iy^ = (2J — iq^y, and therefore 

 x = p^-3pq\ y^ = qUq^-Zp^), 



whence y = qm, provided that 



m2_^3^^^4^ 



Put m-^ipsj'^ = (r + is V 3)*, 



and therefore ^ = 4^35 — 12^5^^ 



7w = r4-18r2s2 4.954^ 



<^.= r2 + 3s2. 

 Whence 



x = 4.rs\ r2 - 35^ | (3r8 + "lOr^s^ + 258;-*5* + 180r2s« + 243s«), 



y = (r^ + 3s2) (r^ - 1 8^252 + 954) ^ 



z = r^ + 28^652 _ 42r*s* + 252r2s6 + sis^. 



Putting 9' = s we find 88^ + 4* = 20^. This, however, is not a 

 satisfactory solution, as it is derived from IP + 2^ = 5^ by 

 multiplying through by 2^'. The solution obtained by putting 

 r=2s is 



571122 + 329* = 2465». 



