414 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



Then (1) has the solution 



Since we may change the sign of P or Q, we get four solutions. If 

 R = A m B n - , where A, B, are expressible in a single way in the form 

 P 2 aQ 2 , it is known that R is expressible in this form in exactly |TT 

 ways when 7r=(m+l)(n+l) is even, and in (7r+l)/2 ways when TT 

 is odd. If a is negative there is only a finite number of solutions of 

 t 2 au 2 = R, since P au 2 = l is not solvable, so that the number of factors 

 A, B, is limited. But if a is positive, let p, q be the least solution of 

 p 2 aq 2 = 1; then every solution is given by 



q = 



, 

 * 



for w = l, 2, 3, . Then 



R = (P 2 - aQ 2 ) (p' 2 - aq' 2 ) = P\-aQ\ 

 if 

 (4) P^Pp^aQq', Q, = Pq'Qp f . 



If we employ as P, Q the various sets corresponding to the factors >1 of 

 the form P au 2 of R and take m = 1, 2, 3, , we get by (4) ail the rational 

 solutions of Pl aQ\ = R. Returning to (3), Lagrange proved that, if 

 the values (3) of x, y which correspond to the case m = are integers, there is 

 an infinitude of values of m (the multiples of an assigned number depending 

 only on a and a) for which the solutions x, y are integers. 



L. Euler 81 gave two methods of finding the general rational solution of 



f(x, y}=Ax 2 +2Bxy+Cif+2Dx+2Ey+F = Q, 

 given one solution x = a, y = b. In f(x, y) f(a, 6) = 0, set 



We get 



Eliminating y by the second of the preceding pair of equations, we get 



:= -at-2b(Bp 2 +Cpq)-2Dp 2 -2Epq, 

 = bt- 2a(Bq 2 +Apq) - 2Dpq - 2Eq 2 , 

 w = Ap 2 +2Bpq+Cq 2 , t = Ap 2 -Cq 2 , 



and hence obtain, when p, q are rational, the most general rational solution 

 of the proposed equation. Integral solutions may be obtained from values 

 of p, q making co= 1 or 2. 

 For the second method, set 



k = B 2 -AC, N=(BD-AE)/k, P = Ax+By+D, Q = y+N. 

 Then 



Af(x, y)^ 



81 Novi Comra. Acad. Petrop., 18, 1773, 185; Comrn. Arith., I, 549-55; Op. Om., (1), III, 297. 



