CHAP. XII] PELL EQUATION, ax 2 +bx+c= D. 393 



A. Boutin 244 noted that, if A is a properly chosen quadratic function of w, 

 x~ Ay"=l are solved completely by an infinitude of polynomials in m, 

 which satisfy certain differential equations of order two. Thus for 



y s -( 



the recurring series 

 = 0, #i = l, , x H 



for even indices solve the first equation, and for odd indices the second. 

 As functions of m, x n and y n satisfy the differential equations 



Similar remarks are made for A=25m 2 14m+2 and for x 2 Ay 2 = l, 

 A = ra 2 -l, rao: 2 +2, m(m 2 +l). 



J. Romero 245 noted that (ny*x) 2 -(nY2nx+A)y*=l if 



A. S. Werebrusow noted that in x 2 Ay 2 = l, A may have the form 

 a 2 m 2 +26m+c if 6 2 a 2 c= 1. 



A. Holm 246 employed the (w+l)th divisor D n when VC is converted 

 into a continued fraction the length of whose period is c. Let p c , q c be the 

 fundamental solution of x z Cy 2 = l. From one solution p n , q n of 

 x 2 Cy-= ( l)"D n we get all the solutions by use of 



where, if c is even, m ranges over all integers, positive, negative or zero; 

 while, if c is odd, m ranges over only the even integers. 



H. Weber 247 treated P Z)w 2 = db4 from the standpoint of quadratic 

 numbers ^(t-\-u^D), where t and u are integers. 



Necessary or sufficient conditions that x~Dy' 2 = 1 be solvable have 

 been noted. 248 



E. B. Escott asked and A. S. Werebrusow 249 replied for what values of 

 a, b, is [a, b, - , a]/[6, c, , 6] integral (cf . Dirichlet's Zahlentheorie 

 p. 49). 



P. F. Teilhet 250 stated and several proved that if is a root of y 2 -3/3 2 = 1, 

 and /3=}=0, then 6/3 2 +l is not a square. Hence n(n-\-l}(n+2)=3A 2 is 

 impossible. 



P. von Schaewen 251 m&def^Ax--\-Bx+C a square in the following cases 

 (in which D = B 2 -4AC): (i) A = n 2 A 1} D = m?D lt A l +D l =^=q*, since 



m< * A L , B + m( l\L , B-mq\ 



244 L'intermediaire des math., 9, 1902, 60-62. 



245 Ibid., p. 182. 



246 Proc. Edinburgh Math. Soc., 21, 1902-3, 163-180. 



247 Archiv Math. Phys., (3), 4, 1903, 201; Algebra, I, 1895, 395-400; ed. 2, 1898, 438-443. 



248 L'intermediaire des math., 10, 1903, 102, 224; 11, 1904, 156-8, 242; 12, 1905, 53-6, 



249-250; 13, 1906, 243-7 (Werebrusow's results are erroneous). 



249 Ibid., 10, 1903, 98; 11, 1904, 154-6. 

 Ibid., 11, 1904, 68-9. 182-4. 



251 Zeitschr. Math. Naturw. Unterricht, 34, 1903, 325-34. Progr. Gym. Glogau, 1906. 



