71 



Certain Diophantine Problems. 



By J. R. Wilton, M.A., D.Sc, 

 Professor of Mathematics in the University of Adelaide. 



[Read Ap.ril 8, 1920.] 



A rapid and obvious way of obtaining the solution in 

 positive integers (1) of the equation 



is to put 



x-^-iy^f'p-if-iqp. 



Equating real and imaginary parts we have 



x = 'p'^ — q'^y y — ^jjq, and 2 = 7^^ + ^-. 



More generally, if 



x^--^y^- = z^z^^ ... 2;; 

 we put 



x + iy = (p^ + iq^) (jh + k^Y • • • fPn + ^9n)'' 

 and equate real and imaginary parts. 



The solution of 



x^ + y- = fa^ + h-^)z'' 

 is obtained by putting 



x-\-iy = (a-\-ih) (p + iq)^, 



or, wh,en n = 2, by means of the identity 



(ax + hy)^ + (hx -ayP = (a^ + h^) fp^ + q^) . 



Exactly similar solutions may be obtained for the equation 

 fx_a,jf + hy^ = z^z^^ . . . z^. 

 Put 



x-ay + iy s,/h = U (p^ jf.iq^^h)r 

 and equate real and imaginary parts. (In this case it is pos- 

 sible to obtain integer solutions if a and h are rational). 

 And similarly the right-hand side may be multiplied by 



A solution of the equation 



x^^ + xj^ + . . . -\-x^- = i(- 

 is readily derived from the trigonometric solution 



(1) Every letter used will denote an integer (not necessarily 

 positive), except that i, as usual, denotes ^^ — i. 



