CHAP. XII] PELL EQUATION, ax 2 +6a;+c= D. 395 



X n having the same F in X 2 n -P n Y* = -1. They and A. H. Bell 258 solved 

 x 2 19?/ 2 = 3 without using the usual convergents. 



Cunningham 259 used known solutions of i/Dx 1 = 1 to factor numbers 

 of the form ?/ 2 +l. 



A. Aubry' 260 give a history and exposition of the Pell equation. 



J. Schroder 261 noted that if PJQ a (a = 1,2, ) are the convergents to 



holds only for k = 2. P. Epstein (p. 310) noted that this result for k = 2 is 

 a case of the known relation between the general solution of x~ Dy~=\ 

 and its least solution. It is also a case of the following theorem. _If 

 D = a~+b, and b is a divisor of 2a, while Z k fNk are the convergents to 

 then 



Several writers 262 discussed the p's for which x 2 (y z I)p 2 = 1 is solvable. 

 A. H. Holmes 263 noted that 41 is the least prime y for which 



A. Holm 264 noted that, if p, q give a particular solution of x 2 Cy i = 

 and r, s one of x 2 Cy' 2 = 1, all positive solutions of the former are given by 



x-yJC=(p-q^C)(r-s^C) n , n = 0, 1, 2, . 

 R. W. D. Christie 265 noted that if we set x = cos 6, y = sin 6, 



l, X 3 = cos 30 = 4 cos 3 0-3 cos e = 4x 3 -3x, -, 

 Y s = sm 30= (4 cos 2 0-1) sin d = 4x-y-y, , 



which give the successive sets of solutions of X*PYl=l if Xi = x, Yi = y 

 is the first set [cf. Wallis, 48 Euler 65 ]. This was verified for any n. 



Christie 266 proved that, if p n , q n are any convergents of pl2ql=l, 



2 tan- 1 -^-tan- 1 -^ =- = 2 tan^-^-- 1 - 



q n +l Pzn+l 4 p n +l 



Christie 267 noted that successive solutions of X 2 pY 2 = l are given by 



X n +i = 2xX n X. n i) Y n -\-i = 2xi n i ni) 



the initial solutions being 1,0; x,y. From a solution of x-6Qli/= 1, 

 one of Z 2 -601F 2 = 1 is found (pp. 54-5). _ _ 



258 Math. Quest. Educ. Times, (2), 8, 1905, 28-30, 58. 



259 Ibid., 83; Mess. Math., 35, 1905-6, 166-185. He noted (p. 183) eight errata in Degen's 101 



table and various errata in Legendre's 88 tables of 1798 and 1830, including A =397 

 (cf. A. Gerardin, 1'intermed. des math., 24, 1917, 57-8). 



260 Mathesis, (3), 5, 1905, 233. 



261 Archiv Math. Phys., (3), 9, 1905, 206-7. 



262 L'intermediaire des math., 13, 1906, 93, 229-230; 14, 1907, 136. 



263 Amer. Math. Monthly, 13, 1906, 191 (148-9 for erroneous solution). 



264 Math. Quest. Educ. Times, (2), 10, 1906, 29. 



265 Ibid., (2), 9, 1906, 111. 



266 Ibid., 52-3. 



267 Ibid., (2), 11, 1907, 39. Cf. p. 96. 



