THEORY OF COMMENSURABLES. 725 



which are equivalent to 



de — {su + tvf + (sf — tiif ; 

 de = {su — tvy + {sv + tuf ; 



and thus the product de is shown to be the sum of two squares in two different 

 ways. 



Cor. The square of the sum of two squares is also the sum of two squares 



For if 



d = s^ + t^ 

 d^^ s* + 2sH^ + t^ 

 = s*-2s^t^ + t* + 4:sH'' 



=:(s2_f2)2 +(2sty 



13. If a number be divisible into two squares prime to each other in two 

 different ways, it is the product of two numbers, each of which is the sum of two 

 squares. 



Let c be the sum of s^ and f, and also the sum of u^ and t?^, that is, let 



then 



s^ — u^ = v^ — t^, or s + u : v + t : : v — t : s — u 



wherefore the ratio s + u : v + t is not in its lowest terms ; that is, s + u and v + t 

 must have a common divisor, which we may suppose to be e , put then s + u^ex, 

 v + t=ey; and then x : y. \ v—t : s—u, so that we may put v—t=fx, s—u—fp, 

 in which /may have the value unit. 



We thus obtain s + u=ex; v + t =fx 



s-u=fy ; v-t=ey 

 whence 



s=l{ex+fy), t=l ifx-ey) 



so that 



s2 + f2=i {e2^2 + 2efxy +fY +foc^ - '^efxy + e'y^ 

 ==l(e'+r)(x^ + y') 



and thus c is the product of two factors, each of which is the sum of two squares. 

 Cor. Hence no prime number can be divided into two squares in more than 

 one way. 



14. If an even number be the sum of two squares, its half is also the sura of 

 two squares ; and conversely. 



If the even number 2n be the sum of two squares 5^ and f, its half n is also 

 the sum of two squares. 



It is evident that the numbers s and t must either be both even or both odd. 

 wherefore ^{s + 1) and ^{s—t) must be integers ; now i(s + tY + \{s—ty' = ^{s"^ + f) 

 — n, therefore n i3 the sum of two squares 



15. If the successive square numbers be divided by any odd prime a, the re- 

 mainders, including zero, recur in groups of a terms; and of the a— 1 terms 



VOL. XXIII. PART III. 9 I 



