CHAP, xii] PELL EQUATION, ax~+bx-\-c=D. 377 



Dirichlet 134 recalled the fact that if T, U are the least positive solutions 

 of t 2 DM* = I, where D is positive and not a square, all positive solutions 

 are given by 



(n = l, 2, ) 



An infinitude of values of u n are divisible by any positive integer S. He 

 proved that, if N is the least n for which u n is divisible by S, the remaining 

 n's are the successive multiples of N. If D' = DS 2 and if T', U r give the 

 least positive solutions of t 2 D'u 2 = l, then N is determined by 



For any prime factor p of S, let v be the least index for which u v is divisible 

 by p and let p 5 be the highest power of p dividing u v . Then, if e is arbitrary, 

 the exponent of the highest power of p dividing u ve is 6+, where e is the 

 exponent of the highest power of p dividing e. Let Vi, Si be the values 

 corresponding to the general prime factor pi of S = Up? and let N be the 

 l.c.m. of Vip^~ &i (i = 1,2, ). When i, o% increase indefinitely, S/N 

 approaches a limit. The application to quadratic forms will be given under 

 that topic. 



C. A. W. Berkhan 135 gave an exposition of the theory of ax 2 +l=y z 

 and a table of solutions for a ^160. 



M. A. Stern 136 applied new theorems on continued fractions to shorten 

 the work of forming an extended table of least solutions of x*Ay* = l. 

 Given the period for one number, we can find an infinitude of numbers the 

 continued fraction for whose square root has a known period. He gave a 

 table showing the manner in which the continued fractions for the square 

 roots of 163 of the numbers < 1000 can be derived from that for 2. 



A. Cayley 137 gave for D<1000, D=5 (mod 8), a table showing the least 

 odd solutions of x 2 Dy 2 = 4, when it is solvable, or, if not, of x 2 Dy 2 = + 4, 

 when the latter is solvable. The computation was made by means of 

 Degen's 101 table ; if in the second line of the entry for D the number 4 does 

 not occur, there is no solution of x 2 Dy 2 = 4; if the rank of the place in 

 which 4 occurs is even, this equation and also x 2 Dy 2 = 4 is solvable; if 

 of odd rank, only x 2 D?/ 2 = 4 is solvable. Also the least solution can be 

 found by means of the series of quotients (in the first line of the entry) by 

 stopping at the number preceding that above 4 and computing the con- 

 tinued fraction determined by this series. From the least solution of 

 T 2 Dv 2 = 4 we get the least solution # = T 2 +2, y = rv, of x 2 D?/ 2 = -j-4, 

 and the least solution Z=(r 3 +3r)/2, F=(r 2 +l>/2, of X 2 -DY 2 =-1. 

 From the least solution of T Z DU 2 = 4 : we get the least solution 

 X =(T 5 -3T)I2, 2/=(!T 2 - 



134 Monatsber. Akad. Wiss. Berlin, 1855, 493-5; Jour, de Math., (2), 1, 1856, 76-9; Jour, fur 



Math., 53, 1857, 127-9; Werke, II, 183-194. 



135 Lehrbuch der Unbestimmten Analytik, HaUe, 2, 1856, 121-193. 



136 Jour, fur Math., 53, 1857, 1-102. 



137 Jour, fur Math., 53, 1857, 369-371; Coll. Math. Papers, IV, 40. Reprinted, Sphinx- 



Oedipe, 5, 1910, 51-3. Errata, Cunningham, 309 p. 59. Extension by Whitford. 302 



