CHAP. XII] PELL EQUATION, ax <2 +bx+c=D. 397 



R. W. D. Christie 277 expressed the solutions of x 1 ^if= 4 in terms of 

 fifth roots of unity. He and others 278 obtained a double infinitude of frac- 

 tional solutions of x~ py- = l from one integral solution: 



y+2V Zpy-p+1 



~^T ) i> -TT- 



E. B. Escott and A. Cunningham 279 factored u &4 in 2ul=( l) n . 

 Christie and Cunningham 280 proved that, if pi 2ql = dbl, 



(Pnpn+i} 2 + (2q n q n+l ) 2 = ql n+l , Pnpn+i - 2q n q n+1 = =fc 1. 



Cunningham 281 applied his 270 method to factor y 6 e in r 2^ = 1, and 

 gave further examples of the factorization of solutions of Pell equations. 



Cunningham 282 noted relations between the solutions of x 2 3y-= 2, 

 2 2 3w 2 = l, in connection with the factorization of (a 6 +276 6 )/(a 2 +36 2 ). 



G. Frattini 283 proved that if D and 2V are polynomials in a, and D is 

 of even degree, and if x*Dy 2 = l has a known solution in polynomials 

 in a, then all solutions of x 2 Dy- = N can be found from one. 



G. Fontene 284 noted that, if a, b give the least positive solution of 



A. LeVy 285 gave another proof of the result proved by Fontene. 275 

 A. Gerardin 286 noted that, if tl dul l, 



. , . 



2n-l, In ^l^n-l ~ tn-2) 



= t n -\-(LU\. 



Each Ukfui is a composite integer. For/f dgl= 1, 



g n = 



G. Ascoli 287 gave an elementary treatment of ax 2 +bx+c = y z . 

 F. Ferrari 288 cited known results leading to a practical method to find 

 all integral solutions of x 2 Dy 2 = l in the solvable cases. 

 W. Kluge 289 proved that for the integral solutions of 



xl - 2kx n y n -yl = ( - l) n p 



277 Math. Quest. Educ. Times, (2), 13, 1908, 35-6. 



278 Ibid., (2), 14, 1908, 56. 



279 IUd., 105-6. 



280 IUd., (2), 15, 1909, 74-75. 



281 Ibid., 95-6; (2), 17, 1910, 64-5. 

 Ibid., (2), 17, 1910, 110-2. 



283 Atti del IV congresso internaz. dei mat., 2, 1909, 178-182. Cf. Frattini. 255 



284 Bull. math. Slementaires, 15, 1909-10, 65. 



285 Ibid., 66. 



286 Sphinx-Oedipe, 5, 1910, 17-29. 



287 Suppl. al Periodico di Mat., 14, 1910-11, 33-8. 



288 Ibid., 69-75. 



289 Verhandlungen der Versammlung deutscher Philologen und Schulmanner, Leipeic, 61, 



1911, 135-7. Unterrichtsblatter Math. Naturwiss., Berlin, 19, 1913, 9-11. 



