CHAP. xii] PELL EQUATION, ax 2 +bx+c=E3. 361 



where the effect of the inequalities is to insure that the X's shall be positive 

 integers making < e,- < Vs. It is proved at length that, if the proposed 

 equation is solvable, we will finally reach a least positive integer ju such 

 that the term E^ is identical with E and such that E^+i = EI, whence 

 E M+V = E,, and also that E m = 1 for a certain m, ^ m ^ ju- Then 

 e m _i equals the greatest integer j8 which is < V5. For brevity, set 



Since E m = 1, /m/ m gives R 2 BS 2 = E, and the general solution is 

 given by 



By actually multiplying together the factors in f m , it is shown that 



R = &l m i-f- l m 2, >0 = tm Ij 



where the I's are derived from the relations (p. 448) 



The notation is at fault if m = 0, when we have R = 1, $ = 0, and if 

 m = 1, when we have E = e = /5, $ = 1. 



Application is made (pp. 454-94) to various numerical equations (13). 

 For Pell's equation (pp. 494-5), we have E = I, whence /3 = e, m = 0, 

 R = i ; s = 0, 



X = $1^+1-*, F = U> r+s^B=(X+Y^Br, 



where n is a positive integer such that np is even or odd according as 

 r 2 Bs 2 = + 1 or 1. For the former, n is arbitrary if ^ is even, but 

 n must be even if /j is odd. Hence if B is any positive number not a square, 

 r 2 5s 2 = + 1 has positive integral solutions. Lagrange noted (pp. 457- 

 461) that Euler's 65 ' 71 method to derive an infinitude of integral solutions 

 of ax 2 + bx + c = y 2 from a given solution does not always lead to all 

 integral solutions unless fractional values of the parameters be used or 

 unless, in y 2 Bx 2 = A, A is a prime. 



Lagrange 75 " investigated the approximation of roots of algebraic equa- 

 tions by continued fractions and proved that the real roots of any quad- 

 ratic equation with rational coefficients can be developed into a periodic 

 continued fraction, and conversely. 



Lagrange 76 derived his preceding formulas for the solution of 



750 M6m. Acad. Berlin, 23, ann6e 1767, 1769; 24, ann6e 1768, 1770; Oeuvres, II, 560-652 

 (especially 603-15). Trait6 de la resolution des Equations numeriques, 1798; ed. 2, 1808, 

 Ch. VI; Oeuvres, VIII, 41-50, 73-131. 



76 Mem. Acad. Berlin, 24, annee 1768, 1770, 236; Oeuvres, II, 662-726. For simplification, 

 see Lagrange. 85 



