410 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xin 



solvable in integers by developing a root of av 2 -\-2bv-\-c = into a continued 

 fraction, admitting negative terms. 



H. Scheffler 60 treated ax 2 2bxycy' 2 = k. We may take x, y relatively 

 prime. Let D = b 2 -\-ac be positive and not a square. Set a = Q , b = P , 

 c = Q-i Develop the root x/y = K=( VZ>+P )/Qo into a continued fraction 

 and let the quotients be a Q , 0,1, . Set 



DPl 



L n Q n i(^ n i r n1) tyn == ~* 



' Z 



Take Q' = k and seek all integers P' , numerically ^k/2, such that D P' 

 s divisible by k. For each such existing P' , develop K' = (^+P' )/k 

 into a continued fraction. There is no solution unless we can assign a 

 common period P T = P' S , Q r = Q', (r+s even) of the two developments. By 

 use of such a common period or a repetition of that period, he obtained a 

 process for finding all relatively prime solutions x, y. 



C. L. A. Kunze 61 treated x s y 3 = xy in four cases. 



J. J. Nejedli 62 assumed that D = b--\-ac>0 in 



ax* = 2bxy +cy 2 +k. 



Set x = aoy+yi. We get a similar equation, apart from the sign of k, 

 (3) Q 1 y* = 2P 1 yy i +ayl-k, P^aao-b, Qi = c-aa 2 +2a b. 



Taking a to be the greatest integer in r=(6+VZ))/a and repeating the 

 process on (3), we can solve the given equation. The process is equivalent 

 to the development of r into a continued fraction. 



S. Realis 63 noted the identity f(x, y} =f(a, jS)/ 2 (A, 5), where 



f(x, y)=ax 2 +bxy+cy z , 

 x=(aa+bp')A 2 +2ci3AB-caB 2 , y= - 



Given the solution /(a, /3) = h, we get another solution of f(x, y} = h if (as 

 is not always the case) solutions of /(A, B) = l can be found. In par- 

 ticular, from solutions /(a, 0) = 1, f(A, B} = d=l, we get new solutions of 

 /O, 2/) = l. 



J. J. Sylvester 64 proved t'ha 1 tfy-+2gxy 2fx 2 = 1 is solvable in integers 

 if A=2/ 2 +gr 2 is a prime and / is odd. Since u 2 Av- = l is solvable, set 

 u-\-l = o-p 2 , ul=Ao-q 2 , where p, q are relatively prime. Then 



p 2 -Aq- = 2/ff = ^Fl or d=2, 



the upper signs being excluded by the form Sn+3 of A. If p 2 Aq- = l, 

 v = 2pq, we write p, q for u, v and p lt q : for p, q and see in like manner that 

 p] Aql = 1 or 2. Finally, we reach ?r 2 A< 2 = 2, where TT and < are odd. 

 Since every prime divisor of 7r 2 +2 is known to have the form ?' 2 +2s 2 , 



60 Jour, fur Math., 45, 1853, 349-369. 



61 Ueber einige Aufg. Dioph. Analysis, Weimar, 1862. 



62 Ein Beitrag zur Auflosung unbest. quad. Gl., Progr. Laibach, 1874. 



63 Nouv. Corresp. Math., 6, 1880, 342-350. 



64 Math. Quest. Educ. Times, 34, 1881, 21-2. 



