CHAP. Xll] PELL EQUATION, ax~+bx-}-c=n. 375 



F. Arndt 125 simplified the solution of x~ Ay*= 1 when A has a square 

 factor. 



J. F. Koenig 126 stated that Jacobi had remarked to him that if 



and C, D are derived from A, B by changing the sign of ^g, then AB-CD, 

 AC-BD, AD-BC equal, respectively 



(a) m*-fgn\ m' 2 -fri 2 , m" 2 -gri f2 , 

 where 



m =a 2 --/6 2 -0c 2 +/0d 2 , w = 2(ad-6c), 



(|8) ro' =a?+fb z -gc z -fgtf, n' = 2(ab-gcd), 



m" = a 2 -/6 2 +0c 2 -/^ 2 , n" = 2(ac -/6d) . 



Given values of m, , n" for which the expressions (a) are unity, Jacobi 

 desired solutions a, , d of (/3). Koenig employed 



and noted that a 2 , -, d 2 are linear functions of 2, while, by computation, 



z = mm'm"fgnn'n". 



He gave a table of values of a, 6, c, cZ for/=2, 3, 5, 6, 7, 0^20, and values 

 of a, -, d giving x *-fgy*=-l for/, < 100, /#< 1000. 



J. B. Luce 127 , to solve x z ny^=^z\ set n = a 2 6, Vw = ad=,7, whence 



Vn = a (w = 2a/6). 



m 2a m 



In the resulting successive convergents, take the numerators and de- 

 nominators as values of , ?/. If m = 2a/b is integral, p 2 nq' 2 = l for 

 p = om+l, g = w. If not integral, seek a square whose product by n leads 

 to an integral value of 2a/b. He gave a table of such square multipliers 

 for n ^ 158. 



F. Arndt 128 was led by investigations on binary quadratic forms to 

 z 2 D?/ 2 = 4, D>0. If .D=0 (mod 4), its roots are x = 2t,y = u, where 



If D^2 or 3 (mod 4) or D=l (mod 8), its roots are x = 2t, y = 2u, where 

 t 2 Du z =l. For the remaining case D^5 (mod 8), he tabulated the 

 least solutions for those values <1005 of D for which the equation 

 x 2 Zh/ 2 =4 has relatively prime solutions, the solutions being for 

 x 2 Dy*= 4 if it is solvable (such a D being marked D*). If x, y give the 

 least posit ve solutions of the latter, X = x 2 +2, Y = xy give the least positive 

 solutions of X 2 DY Z = +4. If the last is solvable in relatively prime num- 

 bers, its least solution is easily deduced from that for x z Dy 2 =l. 



Math. Phys., 12, 1849, 239-243. 

 ^Zerlegung der Gleichung x 2 fgy 2 = 1 in Factoren, Progr., Konigsberg, 1849, 23 pp. 

 Extract in Archiv Math. Phys., 33, 1859, 1-13. Cf. Jacobi. 106 



127 Amer. Jour. Sc. Arts (ed., Silliman), (2), 8, 1849, 55-60. 



128 Archiv Math. Phys., 15, 1850, 467-478. 



