390 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



able, the least solutions of y 2 = ax 2 +l. From a solution of the former we 

 get the solution yi = 2y 2 -\-l, x^ = 2xy of y\ = ax\-\-\. 



A. Hurwitz 220 proved that the positive relatively prime solutions of 

 u' 2 Dv 2 = m, where \m <2VD, are given by the fractions u/v approxi- 

 mating to VZ), where u/v and rfs are said to form a pair of fractions approxi- 

 mating to x if x lies between them and if su rv= 1. 



A. H. Bell 221 found a special solution of x 2 = Ny 2 +l by setting 



x = 1 -\-Nym/n, 



whence y = 2mn/(m 2 Nn 2 ') and asked when the denominator is unity. 

 He treated the case JV = 94 and x 2 61?/ 2 = 1. 



Emma Bortolotti 222 noted that a root of a quadratic equation with dis- 

 criminant A and having as coefficients polynomials in x can be developed 

 into a periodic continued fraction whose elements are linear functions 

 of x if and only if Au 2 v 2 = 1 is solvable in polynomials u, v in x. If A is 

 of odd degree in x, the latter equation is evidently impossible. 



A. Meyer 223 noted that if t 2 Du 2 ! has a fundamental solution T, U, 

 in which U is relatively prime to a divisor DI of D, it has solutions in which 

 u is congruent modulo DI to an arbitrarily given number. 



G. Speckmann 224 employed the identity 



for m = l, and called the resulting solutions of Pell's equation regular if 

 x = 1 and irregular if x 2 is a divisor > 1 of n 2 a 2 2nm. To solve x 2 Dy 2 = M 

 (M 4= square), he sought a square ?? 2 such that M+r? 2 is a square 2 ; then 

 a solution is x = % J rkrf, y = -n, if D = l-f2A;+A; 2 ?7 2 . 



G. Frattini 225 discussed the solution of x 2 Ay 2 I, where A is a poly- 

 nomial in u, especially when A is of degree 2 or 4. 



Ch. de la Vallee Poussin 226 indicated the advantage in using continued 

 fractions in which all but the first quotient are negative integers. 



G. Speckmann 227 noted that the fundamental solutions T, U of t 2 Du 2 = l 

 are T=x+2, U = l, if D = x 2 +4x-\-3; T = 2x+3, U = 2, if D = x 2 +3x+2; 

 etc. He noted identities like 



(no? + m) 3 (n 3 a 6 + 3mn 2 a 3 + 3m 2 ri) a 3 = m 3 . 



A. Palmstrom 228 gave many recursion formulas and relations between 

 sets of solutions of (a-\-2}x 2 (a 2)y z = 4. If x\, y\ are the least positive 



220 Math. Annalen, 44, 1894, 425-7. 



221 Amer. Math. Monthly, 1, 1894, 53-4, 92-4, 169, 239-240. 



222 Rendiconti Circolo Mat. Palermo, 9, 1895, 136-149. 



223 Jour, fur Math., 114, 1895, 240. 



224 Uebcr unbest. Gleichungen, Leipzig and Dresden, 1895. 



225 Giornale di Mat., 33, 1895, 371-8; 34, 1896, 98-109. 



226 Annales Soc. Sc. Bruxelles, 19, 1895, 111. 



227 Archiv Math. Phys., (2), 13, 1895, 327-333; 14, 1896, 443-5. 

 228 Bergens Museums Aarbog for 1896, Bergen, 1897, No. 14, 11 pp. (French). 



