CHAP. Xll] PELL EQUATION, ax 2 +bx-\-c=n. 369 



P. N. C. Egen 102 gave the 121 values of A < 1000 for which x 2 - Ay 2 = - 1 

 is solvable. 



J. L. Wezel 103 proved that if S is the denominator of a complete quotient 

 (VZ+r)/ for the continued fraction for VA, and if pfq is a convergent, 

 then p 2 Aq 2 S. By the periodicity, we ultimately get an S = 1. Thus 

 x 2 Ay 2 =l is solvable for the plus sign, and for the minus sign only if 

 the length of the period is odd. Also x 2 Ay 2 = C is solvable if there oc- 

 curs in the continued fraction for VA a complete quotient of denominator C. 



In the chapter on biquadratic residues in Vol. Ill will be given reports 

 on the paper by G. L. Dirichlet (Jour, fur Math., 3, 1828, 35-69) where he 

 discussed t 2 qu 2 = ps 2 , p and q being primes and p = l (mod 4), and the 

 related pamphlet of 1861 by H. R. Getting. 



J. Baines 104 found values of n for which 



= ra+l. Then 15m 2 +45m+25 = (mr/s5) 2 if m = 5s(9sT2r)/D, where 

 D = r 2 -15s 2 . As by Euler, D = l if (, r) = (l, 4), (8, 31), (63, 244), (496, 

 1924), -, whence n = 6, 86, 401, 5361, -. 



F. T. Poselger 105 treated rz 2 +l = D by continued fractions. 



C. G. J. Jacobi 106 stated that the solutions of x z ay- = l can be expressed 

 in terms of the sine and cosine of 2w7r/a, and stated that he possessed a 

 generalization to the case in which a is a product of several factors. If 

 a = be, we can find in an infinitude of ways four integers u, v, w, x such that 

 the product of the four factors My V&w Vcz V&c is unity, where two 

 or four of the signs are plus. The resulting relation can easily be given 

 the three forms y z bz z = l, y\cz\ = l,ylazl = l. Hence the solutions 

 y, , 22 depend on u, v, w, x. The latter can be expressed by trigonometric 

 functions. 



T. L. Pistor 107 gave an exposition, illustrated by examples, of the methods 

 of Gauss and Legendre to reduce the general quadratic equation in x, y 

 to v 2 Dy 2 = N, its solution by continued fractions if D > and by trial 

 if D=d, using y = Q, 1, 2, , up to WJd. On p. 44 is given a 

 table of the least solution of Pell's equation x 2 Dy 2 = l, D = 2, - - -, 200. 



G. L. Dirichlet 108 recalled Legendre's 88 result that if p, q are the least 

 positive integral solutions of p z -Aq z = l, then 2=fMg 2 -fNh 2 , where /=! 

 or 2, and MN = A is a decomposition of A. Dirichlet proved that at 

 most one of the latter equations, in addition to l = g 2 Ah 2 , is solvable. 

 Besides Legendre's theorems for primes A = 4n+l, 8w+3, Sn l, Dirichlet 



102 Handbuch der aUgemeinen Arith., Berlin, 1819-20; ed. 2, I, 1833, 457; II, 1834, 467; 

 ed. 3, I, 1846, 456; II, 1849, 468. Cf. Seeling. 148 



103 Annales Acad. Leodiensis, Liege, 1821-2, 24-30. 



104 The Gentleman's Diary, or Math. Repository, London, 1831, 38, Quest. 1268. 

 106 Abh. Akad. Wiss. Berlin (Math.), 1832, 1. 



106 Letter to Legendre, May 27, 1832; Werke, I, 458; Jour, fur Math., 80, 1875, 276; Ann. de 



PEcole Normale Sup., 6, 1869, 176-7; Bull. Sc. Math. Astr., 9, 1875, 139. Cf. Koenig. 126 



107 Uber die Auflosung der unbest. Gl. 2. Grades in ganzen Zahlen, Progr., Hamm, 1833. 

 is Abh. Akad. Wiss. Berlin, 1834, 649-664; Werke, I, 219-236. 



25 



