CHAP, xill] Ax-+2Bxy+Cy*+2Dx+2Ey+F = Q. 415 



Let G and // be the values of P and Q for x = a, y = b. Then 



Equate the first factor on the left to the second factor on the right and 

 vice versa. Thus 



. 



A. 



Or, use the Pell equation s 2 fcr 2 = l, having an infinitude of solutions if k 

 is neither negative nor a square, and set 



By equating the terms free of V/fc, we get rational expressions for x, y. 

 Euler 82 treated the solution in integers of 



(5) az 2 +/3z+7 = ft/ 2 +W+0, 



given one solution x = a, y = b. Denote the roots of 2 2 = 2sz 1 by 



p = s+V s 2 -l, q = s- Vs 2 -l. 

 Make the substitution 



(f\\ r Q 



(6) X 



Since p<? = l, the members of (5) equal respectively 



/2p 2 + n +2/sr+7 ._l 



These are equal if 



j8 2 r? 2 

 (7) Vg ,!L-i. 



For w = 0, let x = a, y = b. Then (6) gives 



and the resulting value of (f+g) 2 (fgY~ reduces to (7) since (5) holds for 

 x = a, y = b. Hence the values of/, g from (8) lead to solutions (6) of (5) 

 provided s, in the expressions for p and q, is such that the resulting x, y 

 are rational. For n = 1, the expressions for x, y become, in view of (8), 



s 2 -! 



Then s 2 = 1+afr 2 , a solvable Pell equation if f is positive and not a square. 

 Hence if the latter is solved and we set p, q = sr^ and define /, g by 

 (8), then, for any integer n, (6) gives a solution, which is proved rational 

 as follows. Call x', y' the values obtained from (6) by changing n to n+1 ; 

 x", y" those by changing n to n+2. Then 



(s-l), 2/ // = 2 S2 /'- 2 /+^( s -l). 

 _ ; __ T _ 



K Mm. Acad. Sc. St. Petersb., 4, 1811 (1778), 3; Comm. Arith., II, 263. 



