CHAP, xii] PELL EQUATION, ax 2 +bx+c=t3. 355 



2qB A, -. Hence if one solution ap 2 + 1 = q- is known, we get an 

 infinitude of solutions p' = 2pq, etc. Euler noted special forms of numbers 

 a for which a solution of ap 2 -{- I = q 2 may be given at once, viz., (a, p, q) : 



i 



e 2 - 1, 1, e; e 2 + 1, 2e, 2e 2 + 1; a 2 e 2b d= 2ae b ~\ e, ae b+l 1; 

 (ae b + /3e") 2 + 2ae b ~ 1 + 2^~\ e, ae b+1 

 iW 6 d= ae b ~\ ke, iak 2 e b+1 1. 



If a is not of one of these forms, apply the method explained by Wallis, 

 which is here illustrated for 31p 2 + 1 = q 2 . Euler gave a table showing, 

 for each a ^ 68 not a square, the least positive integer p and the corre- 

 sponding q satisfying ap 2 + 1 = q 2 . From Va = V 2 1/p, Euler noted 

 that, if q is sufficiently large, q/p is a close approximation to Va; let P 

 be the iih term of the above series 0, p, 2pq, and Q the ^th term of the 

 series 1, q, 2q 2 1, such that aP 2 + 1 = Q 2 ; then the successive 

 values of Q/P are closer and closer approximations to Va. 



Euler 66 noted that the least integral solution x of ax 2 + 1 = D is 

 226153980 for a = 61, and 15140424455100 for a = 109, and stated he 

 could shorten very much the work necessary by "Pell's method." If 

 x 2 ey 2 = N has the solution a, b, it has also the solution 



2ebq - 2ap 

 x -- a + pz, y ----b + qz, ^ _ ^ . 



Making use of the existence of integral solutions of p 2 eq 2 = 1 for e not a 



square, we can assign an infinitude of integral solutions of x 2 ey 2 = N, 



since 



(11) N = (a 2 - eb 2 )(p 2 - eq 2 } = (ap ebq) 2 -- e(bp aq} 2 . 



This formula of composition was known 67 by Brahmegupta. 28 



R. Simpson 68 noted that if we are given a and a fraction b/c such that 

 (b 2 =F l)/c 2 = a, the series of fractions, converging to 



b d b 2 -\- ace / bd + ace h bf + acg 

 c' e~~ 2bc ~' g~~ cd + be ' k ~~ cf + bg ' 



are such that the numerator of any fraction (as hfk) is the sum of the prod- 

 ucts of the numerators and the denominators of b/(ac) and the preceding 

 fraction (then fig), while the denominator (then 7c) is the sum of the prod- 

 ucts of the numerators and denominators of c/b and that preceding fraction 

 (fig). By (11), every fraction N/D in the series has the property 

 N 2 - 1 = aD 2 if b 2 - 1 = ac 2 ; but if 6 2 + 1 = ac 2 that property holds 

 only for alternate fractions, while N' 2 -}- 1 = aD 2 for the others. He cited 

 the " obscure passage " where A. Girard 68a gave the approximations 577/408 



66 Corresp. Math. Phys. (ed., Fuss), 1, 1843, 616-7, 629-631; letters to Goldbach, Aug. 4, 

 1753, Aug. 23, 1755. 



67 Cf. M. Chasles, Jour, de Math., 2, 1837, 37-50. Reprinted, Sphinx-Oedipe, 5, 1910, 65-75. 



68 Phil. Trans. London, 48, I, 1753, 370-7; abr. ed., 10, 1809, 430-4. 



68a Les Oeuvres math, de Simon Stevin de Bruges . . . par A. Girard, Leyde, 1634, I, 170. 



