CHAP. XII] PELL EQUATION, ax-+bx-\-c= D. 379 



M. A. Stern 145 proved (p. 27) that x 1 Aif = d n has one and but one 

 solution in integers if d n (0<d n < VA) is the denominator of a complete 

 quotient which belongs to a partial denominator a n+ i of the " negative ' 

 periodic continued fraction 



r I 1 l 

 \_a, i, a 2 , j = a 



di a^ - 



for VA, and x, y are the numerator and denominator of the convergent 

 [a, i, , a n ]. The first d n which is unity leads to the solution of 

 z 2 Ay 2 = l in least integers; this d n is the denominator belonging to the 

 final term of the first period. He found (pp. 30-43) the conditions for 

 d TO = 2. Finally there is a table giving for A^<100 the partial denominators 

 of the half period and the complete quotients for the negative continued 

 fraction for VA 7 ". Lagrange 85 had shown by an example that Pell's equation 

 cannot be solved by use of a continued fraction in which the partial de- 

 nominators have signs chosen at will. 



J. Frischauf 146 noted that Gauss 93 (Arts. 197-202) obtained the least 

 solutions T, U of P Du-=ff 2 by use of a reduced quadratic form of deter- 

 minant D. It is here shown that T, U are independent of the particular 

 reduced form used. 



N. de Khanikof 147 used a table showing the last two digits of the root 

 of a square ending in 01, 04, , 96 to find the endings of possible integral 

 solutions of A+Bt 2 = u 2 . 



P. Seeling 148 treated the form of numbers the continued fractions for 

 whose square roots have periods with a given number g of the terms, 

 treated the cases g = 2, , 7 in detail, and tabulated the period of the 

 continued fraction for VA, 2^A^602. He noted that Egen 102 omitted 

 from his table all numbers of the form n 2 +l, though they belong there. 

 Egen stated that x~ Ay 2 = 1 is solvable only when the period of the con- 

 tinued fraction for VA has an odd number of quotients. Seeling stated 

 that it is possible in relatively prime integers x, y only when A=4ra+l or 

 4ra-{-2. Hence if the period for VA has an odd number g of quotients, 

 A=4ra+l or 4w+2; this is proved for g = l, 3, 5, 7. 



L. Ottinger 149 gave tables showing several solutions of x z Ay 2 =b 

 for A = 2, ,20; 6 = 1, , 10, 3*, 5 fc , 7 k ( = 1,2,3,4). If we have found 

 by continued fractions the least solution of p 2 A<? 2 =6 and know a 

 solution of t 2 Ai* 2 = l or 1, another solution of x 2 Ay 2 = 6 is given by 

 a=ptAgu, y = puqt. 



A. Meyer 150 proved by use of ternary forms that if D is a positive integer, 

 2" the highest power of 2 dividing D, cr^4, S 2 the greatest odd square 

 dividing D, and D = 2'S 2 Di, then there exist integers , 77, relatively prime 



145 Abhand. Gesell. Wiss. Gottingen (Math.), 12, 1866, 48 pp. 



146 Sitzungsber. Akad. Wiss. Wien (Math.), 55, II, 1867, 121. 



147 Comptes Rendus Paris, 69, 1869, 185-8. 



148 Archiv Math. Phys., 49, 1869, 4-44. 



149 Ibid., 193-222. 



1BO Diss., Zurich, 1871; Vierteljahrsschrift Naturf. Gesell. Zurich, 32, 1887, 363-382. Cf. 

 Got. 299 



