CHAP, xill] SOLUTION OF ax~+bxy+cy z = k. 411 



TT V-2= (gr+jfV^Xy+z >/^2) 2 . By the coefficients of V-2, 



S. Roberts used reduced quadratic forms and results of A. Gopel. 



E. Cesaro 65 proved that the number of sets of positive integral solutions of 



Ax 2 +Bxy+Cy z = n (A>0, C>0) 



is Tr/(25)-B/d 2 in mean, where 6 2 = 4AC- 2 . 



S. Realis 66 noted that if a, ft is a solution of x-+nxy ny* = l then 

 x = (n-\-l)a n/3, y = (n+2)a (n+l)/3 is a solution. From the evident 

 solution 1, 0, we get the solution n+1, n-f-2. Using y = n+2, and solving 

 the initial equation we get x = n-\-l and the new value x n 2 3n 1. 

 Applying the formula to the latter we get a fourth solution, etc. The ath 

 set x a , y a of solutions of this series is given, as well as recursion formulae. 



Realis 67 noted that mx~ (m+nl')xy-{-ny~ = h has the solution 



(4) x = (m ri)a (m nl)j3, y=(m 7iTl)a: (m n)/3, 



if a, |8 is one solution. Starting from this set (4), we get again the first 

 set a, 13. Evidently (4) hold also for an equation derived from the given 

 one by increasing m and n by the same number; also for 



For x- (n-}-2}xy-{-ny 2 = l, the solution 1, gives the solution n 1, n. 

 For y = n, we have x = n l, n 2 +n+l, and hence find an infinitude of 

 solutions. There is treated the equation obtained from the last by changing 

 the sign of the constant term, and 



x*-2(n+l)xy+(2n-l)y* = l or -2. 



Recursion formulae are given for the integral solutions of x 2 Axy-\-By" = h 

 when A 2 is divisible by A B 1 . 



*P. S. NasimorT 68 gave an exposition of Jacobi's series for elliptic func- 

 tions and application to the number of solutions of ax 1 -\-~bxy -\-cy' 1 = n, in 

 particular for x 2 -\-16y 2 = n, 4x 2 +4x?/+3?/ 2 = n, ax*+cy z = n (a, c odd). 



F. J. Studnicka 69 noted that if p k and q k are the numerator and de- 

 nominator of the &th convergent for the continued fraction 



Using qn^aqn-i+qn-z, we get 



aq n -zq n 

 and hence the solutions of axy+x*y~ = l. Cf. Kluge 289 of Ch. XII. 



65 Mem. Soc. R. Sc. de Liege, (2), 10, 1883, No. 6, 197-9. 



66 Nouv. Ann. Math., (3), 2, 1883, 494-7. 



67 Ibid., (3), 3, 1884, 305-15. Errata, p. 448. 



68 Application of elliptic functions to the theory of numbers, Moscow, 1885, Ch. 1. French 



resume in Annales sc. de 1'ecole normale super., (3), 5, 1888, 23-31. 



69 Prag Sitzungsber. (Math. Nat.), 1888, 92-95. 



