CHAP, xii] PELL EQUATION, ax 2 + bx +c= D. 357 



a be the largest integer ^ ( \k + v)/a. In x = a + I/?/, 



1 a _ a( V^ V-\- aa) 



~x--a~ Vz + y-aa:~ z--(v--aa) z 



where B = aa v, /3=1 + 2av a 2 a. Hence let 6 be the largest integer 

 ^ (Vg + J5)/|S. Taking y = b+ 1/t, and proceeding similarly, we obtain 

 Euler's table: 



_ 

 I A = v, a = z A- = z v 2 , a ^ 



OL 



z 7? 2 w -4- Ft 



B = aa- A, |8 = 



III C = j86-B, 7 = 



, 



a: p 



?j -t- r' 



P 



IV D = 7C - C, 5 = = + C (C - 



7 o 



etc., where in the last column the equality sign is taken only when the 

 fraction is an integer. It follows that A, B, C, are ^ v, and the indices 

 a, b, c, - are ^ 2v. Euler observed in many examples that when the 

 value 2v is reached, the values a,b,c, repeat; but no proof 73 is given 

 that the index 2v exists [proof by Lagrange 74 ]. For each z ^ 120 and 

 not a square, he gave the values of v, a,b,c, (at least as far as a period), 

 and underneath them the values of 1, a, f3, 7, . Such values are given 

 also for certain types of numbers, viz., z = n 2 + k, k = 1,2, n, 2n 1, 2n, 

 and z = 4n 2 + 4, 9n 2 + 3, 9n 2 + 6. 



The successive convergents v, (va + l)/a, to Vs are found by the law: 



v, a, b, c, , m, n, -, 



1 v av+ 1 (ab + l}v + b M N nN + M 



6' 1' ~^~ a6 + l '' P' Q' wQ + P' 



These convergents are given the symbolic notation 



1 0) (v, a) (v, a, b) (v, a, b, c) 

 6' T' 7^~' (a, 6) ' (a, 6, c) ' 

 where 



(v) = v, (v, a) = v(a) + 1, 0, a, 6) = v(a, 6) + &, 



(v, a, b, c) = v(a, b, c) + (b, c), 

 He stated that 



(v, a, b, c, d, e) = v(a, b, c, d, e) + (6, c, d, e) (v, a)(b, c, d, e) + f(c, d, e) 

 _ = (v, a, 5) (c, d, e) + (v, a) (d, e} = (v, a, b, c) (d, e) + (v, a, b) (e), 



As remarked by H. J. S. Smith, British Assoc. Report, 1861, 96, pp. 313-5; Coll. Math. 

 Papers, I, 1894, 194, Euler's paper contains all the elements necessary to give a rigorous 

 proof of this fact and hence that the process always leads to a solution, other than 

 x = I, y = 0, of z 2 zy 2 = 1. Plana 69 noted that Euler's proof becomes rigorous if 

 slightly modified as by Legendre. 87 For (a, j8, )> Gauss 24 of Ch. II wrote [a, /S, ] 



74 Miscellanea Taurinensia, 4, 1766-9, 41; Oeuvres, 1, 1867, 671-731. 



