CHAP. XXII] 2z 4 -7/ 4 =D; SYSTEM x+y = A z , x-+y- = B 4 . G23 



J. L. Lagrange 54 discussed Format's 37 problem at length. From 

 P+q y~> p 2 +q~ = x 4 , he derived, after setting z = p q, 



(2) 2x l -y* = z 2 . 



The problem reduces to the solution of (2) since we have 



(3) P = i(0 s +*), <? = f(2/ 2 -z). 



Lagrange was evidently not acquainted with Euler's 52 paper of 1773 in 

 which he derived (2) and obtained four sets of solutions A = x, B = y, 

 indeed, Lagrange omitted the set 1525, 1343, in his citation of Euler. 

 Given any integers x, y for which 2z 4 if= D, Lagrange gave a method to 

 obtain smaller integral solutions ; then by reversing the process and starting 

 with x = y = l, he concluded that all pairs of larger solutions can be found 

 in the order of their magnitude. While Euler's simpler procedure appears 

 to give all the solutions in this manner, he did not prove that this is the case. 

 We may assume that x and y are relatively prime. A simple argument 

 shows that x, y, z are all odd. By (2), 



Denote these (even) factors by 2mp, 2mq, where p and q are relatively prime. 

 Then pq must be a square. Hence, replacing p, q by p 2 , q 2 , 



Eliminating z from the first two, by means of the third, we get 



Thus m = 1 or 2, since m is a divisor of 2x 2 and 2y~. If ra = 2, set p+q = q' f 

 Q.~P p'' Whether m = l or 2, we obtain equations of the form 



(4) x 2 -y 2 = p(p-q), x 2 +y 2 = q(p+q}. 



Thus p is odd. Set (x+y)lp = 2mfn, where n is odd and prime to m. Then 

 x+y = 2ms, p = ns, where s is an integer. By (4i), xy = 2nt, p q = 4mt, 

 where t is an integer prime to s. Thus 



(5) x = ms+nt, y = msnt, p = ns, 

 Then the product of (4 2 ) by (s 2 -8 2 )/n 2 gives 



77? 



s 4 +W = u 2 , u=3st+-(s* 



n 



Since m and n are relatively prime we therefore have 



(6) m = (u-3s)/l, n = (s 2 -St-)/l (I an integer). 



If m = 0, then s/t=l,ri* = I,x 2 = y 2 = l. Hence if (2) has a set of relatively 

 prime solutions x, y not both of numerical value unity, then by (5) the 

 greater of x, y exceeds the greater of the corresponding solutions s, t of 



(7) s 4 +8 4 = w 2 , 



and s, t are relatively prime and not both of absolute value unity. Con- 

 versely, from relatively prime solutions s, t, we obtain by (6) and (5) rela- 

 tively prime solutions x, y of (2) . 



" Nouv. Mem. Acad. Sc. Berlin, annde 1777 [1779], 140; Oeuvres, 4, 1869, 377-98. 



