CHAP. Xii] PELL EQUATION, ax 2 +bx+c=\U. 367 



John Leslie 90 treated x 2 -\-y 2 -\-bxy = a 2 by factoring a 2 y 2 , solved 



Ax 2 +Bx+C = y 2 



if A, C or B 2 4AC is a square, and derived a second solution of ax 2 -\-b = y z 

 from one solution. 



P. Paoli 91 noted that, if t = h, u = k give one set of rational solutions of 

 Ai--\-B = u 2 , all are given by 



hr 2 -2kr+Ah 

 t = -7 1 , u = k+r(t-li). 



P. Cossali 92 discussed Euler's and Lagrange's methods to solve (13). 



C. F. Gauss 93 showed how to find all solutions of t 2 Du 2 = m 2 , given 

 two linear substitutions which transform any reduced form AX 2 -\-2BXY 

 + CT 2 of determinant D into the same quadratic form (see quadratic 

 forms in Vol. III). 



J. C. L. Hellwig 94 gave an exposition of Pell's and other equations of 

 degree 2. 



R. Adrain 95 reproduced the simpler proofs from Euler's 83 Algebra, II, 

 Chs. 4-5. 



F. Pezzi 96 employed the continued fraction 



x = a,-\ -= 



where M n /N n is the convergent derived by deleting l/x n . Take x= VZ, 

 jCi=l/(VA a), etc. Then x n = (ifA+b n )/Cn } where 



By substituting this value of x n and the corresponding value of x n+l in 

 x n = ci n -\-l/Xn+i and equating rationals and irrationals, and changing n to 

 n 1, we get 



Since the a's do not exceed 2a, the a's repeat after a certain number n of 

 terms. Then Ml = AN* + ( l) n . Hence x 2 -- Aif = 1 is solvable in 

 an infinitude of ways, likewise x 2 Ay 2 =l if and only if the period 

 length n is odd. Consider any solutions of M* m =AN z m +(V) m . If N m is 

 even, M m is odd and m even. If A is even and N m odd, M m is odd and 

 (!) = ( l) m . If A and N m are odd, N OT is even and (-1)"= ( 1)' B+1 . 



90 Trans. Roy. Soc. Edinburgh, 2, 1790, 193-209. Reprinted in the Math. Repository (ed., 



Leybourn), London, 1, 1799, 364; 2, 1801, 17; Encycl. Britannica. Cf. Berkhan. 135 



91 Element! d'algebra, Pisa, 1, 1794, 165-6. 



92 Origine, trasporto in Italia . . . Algebra, Parma, 1, 1797, 146-155. 



93 Disquisitiones Arithmeticae, 1801, arts. 162, 198-202; Werke, 1, 1863, 129, 187; German 



transl. by Maser, 1889, 120, 177-87. Cf. Dirichlet. 133 



94 Anfangsgrtinde der Unbest. Analytik, Braunschweig, 1803, 80-184. 



95 The Math. Correspondent, New York, 1, 1804, 212-222 (first American math, periodical). 



96 Memorie di Mat. e di Fisica Soc. Ital. Sc., Modena, 13, 1807, I, 342-365. 



