CHAP. XII] PELL EQUATION, ax-+bx+c= D. 389 



E. Lemoine 215 proved that all positive solutions of x 2 +l = 2?/ 2 are given 

 by x n = Nzn-i+Nzn, Vn = N Zn , where Nn-td+Nnb is the nth term of the series 

 Ui = a, u z = b, -, u n = 2u n -i-\-u n -z, so that 



If z, ?/ is a solution of x*-\-l = 2if, then x+2t/, x+y is a solution of x 2 l = 2y 2 

 and the same holds if the equations are interchanged. 



G. B. Mathews, 216 employing the fundamental solution (T, U) of 

 t 2 Du 2 =cr 2 , and the notation of hyperbolic functions, put 



</> = cosh- 1 (7 7 /(r)=sinh- 1 (t/VI)/<r). 



Then the general solution is T n = a- cosh n<f>, U n = (<rf V.D) sinh n<. 



K. Schwering 217 started with Jacobi's elliptic function x = sin amw, the 

 function inverse to 



u= I dx/^l-x*, 



t^O 



and an "odd" integral complex number rj = a-\-bi, where a is odd and 6 

 even, so that q = a 2 +6 2 is odd. Then 



_q 1 

 



_ 



sin am (TIU) = - 77 -7-7 : = / 4N > v 



9 ~ 4 



If 77 is a complex prime of the form 4/c+3+(4fc / H-2)i, then <}>(x*}, on which 

 depends the division of the lemniscate by 77, is factorable into 



Let g be a primitive root of the prime q, so that g"=i (mod 77). Taking 

 x=l, we get odd complex integral solutions t, u of t 2 r]U? = 2i( l) ln '. 

 By squaring <+ A/i?w we get complex integral solutions of T 2 rjU 2 =l. 



H. Weber 218 employed the modular equation (an algebraic equation in u 

 and v of degree 24 in each) which holds between the two elliptic functions 

 M=/(CO), y=/(23w), to deduce the identity X 2 M-7W = 1, 



Squaring ZVM+F^, we get x+y^D, where z 2 -Zty 2 = l, D = MN. 



C. E. Bickmore 219 computed (for a committee of which A. Cayley was 

 chairman) a table, extending Degen's 101 and showing, for 1001 ^a^ 1500, 

 the least solutions of y z = ax 2 1 when a is not of the form Z 2 +l (in the 

 contrary case, y = t, x = l, give a solution), and, when the latter is not solv- 



215 Jornal de Sc. Math, e Astr. (ed., Teixeira), 11, 1892, 68-76, 115. 



216 Theory of Numbers, 1892, 93. 



217 Jour, fur Math., 110, 1892, 63-4 (112, 1S93, 37-8). 



218 Math. Annalen, 43, 1893, 185-196. 



219 Report British Assoc. for 1893, 1894, 73-120; Cayley's Coll. Math. Papers, 13, 1897, 430- 



467. Errata by Cunningham. 256 - 309 



