CHAP, xxii] 2x 4 -7/ 4 =D; SYSTEM x+y = A 2 , x 2 +y 2 = B*. 625 



we have A 2 2B 2 = (t 2 2u~) 2 . Noting that a solution involving fractions 

 may be replaced by an integral solution, he took s = l, whence r tu. 

 Equating the two expressions for A, we get 



For u = l, 2 = 3/2. The latter leads to the second value u= 13, which in 

 turn gives t= -113/84. Then u = 301993/1343, etc. Euler stated that 

 it is easy to see that the pairs of adjacent values of u, t give all sets of 

 rational solutions. From the formulas for the sum of the roots of a quad- 

 ratic equation, we see that 



,, 22 ., . , 2u' 



u'+u= -, i 



if u, t, u f , t' are consecutive terms of the series. 



Euler, in the third paper, set A/B = (1 +x)/(l x) in 2A* - B* = D . Thus 



In accord with his 143 general method, he set l+6z+z 2 =X(p 2 +8<? 2 ), x = \pq. 

 Cf . Euler, 143 end. 



V. A. Lebesgue 56 gave a method simpler than Lagrange's (whose article 

 he had apparently not seen) to obtain from given solutions of (2) a smaller 

 set of solutions. Since p 2 -\-q 2 = x*, we may set p = 2mn, q = m 2 n 2 , 

 x z = m 2 +n 2 , where n is even since p-\-q is a square y 2 . By the third relation, 

 m = r 2 s 2 , n = 2rs, rc = r 2 +s 2 , where one of the integers r, s is even and the 

 other odd. Changing the sign of y if necessary, we may assume that, of 

 the factors r 2 -\-2rs s 2 y of 8rV (in view of p+q = y*), the one with the 

 upper sign is divisible by 2 but not by 4. For r odd we may therefore set 



r 2 +2rs-s 2 -j-2/ = 2-r 2 r 2 +2rs-s 2 -y = 4:^s 2 , 



u t 



where u, t are odd and relatively prime. Multiplying the sum by \ut, we get 

 (8) r 2 (t 2 -ut) -2rsut+s 2 (2u 2 +ut)=0. 



For s odd, the right members are obtained by interchanging r, s, and the 

 new sum is derived from (8) by replacing r/s by s/r, and t by t. By (8) 



s 



Since ut and 2u 2 t 2 are relatively prime, each is a square or the negative 

 of a square. But t and u are odd, and t 2 2u 2 is of the form 8k 1 and not 

 a square. Hence, taking u and t positive, we may set u =/ 2 , t = g 2 , 2f 4 g* = h z . 

 Then 



s g 



If x, y do not have a common square factor, r, s are relatively prime and 

 or=/A, <rs = gB, where <r is prime to / and g. Then y = rHfu 2s 2 u/t and 



- 2/ 2 5 2 , a 2 x 2 =f 2 A 2 +g 2 B 2 , <r*z = C 2 -2(f 2 A 2 - g 2 B 2 ) 2 , 



68 Jour, de Math., 18, 1853, 73-86. Reprinted, Sphinx-Oedipe, 6, 1911, 133-8. 

 41 



