CHAP. XII] PELL EQUATION, ax~+bx+c= D. 391 



solutions of x- Ay~= 1, then 



so that y/iji, z/#i have the same properties as the above x, y, where now 

 a = 4zi+2. If Xi is the least positive integer for which x 2 Aif= 1, we see 

 that, by taking a = 4z? 2, the solutions of odd rank have the same proper- 

 ties as the solutions of x~ Ay z = I. 



C. Stormer 229 quoted the known result that, if a, b are the least positive 

 solutions of x 2 Ay 2 = 1, other solutions are given by 



__ 



and solutions of x 1 Ay- = +1 are given by x 2n -\-y 2n ^A = (a+6 A/A) 271 . He 

 proved that 



a-/3 = 2 tan" 1 --, a + = 2 tan" 1 --, 



Stormer 230 noted that if x z Dif= d=l (-D>0) has positive integral solu- 

 tions and ?/i is the least y, either there is no solution y such that every 

 prime divisor of y divides also D, or there is only one such solution, viz., y. 



A. Thue 231 proved that in x 2 Dy 2 = m the least positive y is ^^Vw, 

 where y is a positive number for which u 2 Dv 2 = l, provided D is not a 

 square and u>l. 



A. Boutin 232 tabulated the periods of continued fractions for Vn, 

 w^200, and when n is one of 30 special quadratic functions of a parameter 

 [cf. Stern 109 ]. He 233 gave the complete solution of y 2 (m 2 I)x 2 =l, with 

 details when m = 2. 



H. Brocard 234 gave references to problems depending on x z 2y 2 =l. 



E. de Jonquieres 235 proved by the use of binary quadratic forms that 

 (a 2 -4)z 2 -4i/ 2 =l is not solvable if a + 3, that (a 2 -l)o: 2 -4?/ 2 =dbl is 

 not solvable [error for 1], that (a+l)z 2 ay 2 = l (a>0) has the least 

 solutions = 4a+l, y = 4a-\-3, that (ma 2 l)x 2 my 2 =l has the least 

 solutions = 4wa 2 dbl, 2/ = 4ma 3 d=3a, and gave long expressions for solutions 

 of (wa 2 d=4)x 2 my 2 = 1 (a and m odd). The method employed is similar 

 to that of Gauss (Disq. Arith., art. 195), but with the variation (inspired 

 by Legendre) that he omitted from the period of neighboring reduced 

 forms those having the middle term zero. He applied (pp. 1077-81, 1177) 

 Gauss' method of reduction to (ma 2 +4) 2 my 2 = l. He gave (p. 1837) 

 values of D for which t 2 Du 2 =l is solvable in integers: D = a 2 (n 2 +l), 

 D = 4n 2 +4n+5, where a is a divisor of any term of odd rank in the recurr- 

 ing series having 0, 1, 2n as initial terms and having 2n, 1 as the scale of 

 relation. It is not solvable if D = a 2 (n 2 +l), n a multiple of a. 



229 Nyt Tidsskrift for Math., 7, B, 1896, 49. 



230 Videnskabs-Selskabets Skrifter, Christiania, 1897, No. 2, 48 pp. Cf. Stormer. 274 



231 Archiv for Math, og Naturvidenskab, 19, 1897, No. 4. 



232 Mathesis, (2), 7, 1897, 8-13. 



233 Ibid., (2), 8, 1898, 159-161. 



234 Ibid., 112-3. 



235 Comptes Rendus Paris, 126, 1898, 863-871, 991-7 (correction, 132, 1901, 750, and 1'inter- 



mediaire des math., 8, 1901, 108). 



