CHAP, xii] PELL EQUATION, ax-+bx+c= D. 371 



G. L. Dirichlet 113 solved t 2 pu 1 = 1 by use of trigonometric functions and 

 remarked that the method is not so well adapted to numerical calculation 

 as that by continued fractions and does not give the least positive solutions. 

 Let GI, , a s be the s=(p l}/2 quadratic residues of the odd prime p, 

 and let 61, , b s be the quadratic non-residues. Write i= V ~L In 



where the upper or lower sign is taken according as p = 4/^+1 or 4^+3, Y 

 and Z are polynomials in x whose coefficients (as shown by Gauss, Disq. 

 Arith., art. 357) are integers. By multiplication, we get 



fp i 

 X=- . 



x 



Let p = 4/*+l. For x = l, let Y, Z become the integers g, h. Then 

 g 2 ph 2 = 4p. Hence g = pk, Ji 2 pk" = 4. It remains to evaluate g and 

 h. Since ai, , a s have in some order the same remainders as I 2 , 2 2 , , s 2 

 when divided by p, we have 



3=1 3=1 



since 



l+2 2 + . 



^4 



(mod 2). 



In terms of the trigonometric product a, we evidently have 



a 2 1 /a 2 



To pass from these solutions of h 2 pk 2 = 4 to solutions of t- pu 2 = l, let 

 first p = 8v+l; then h and A; are both even, so that 



h k 



For p = 8j'H-5, it is stated that /i and & are both odd, whence solutions t, u 

 are easily deduced. But R. Dedekind 114 noted that both h and k can be 

 even, as for p = 37, 101, etc. Finally, if p = 4ju+3, it is shown that, for 

 x = i, Y and Z become g(\i] and /i(lTi), where g and /i are real integers, 

 and the upper or lower sign holds according as p=7 or 3 (mod 8). Evi- 

 dently X becomes i. Hence Y 2 +pZ 2 = 4X takes a form equivalent to 

 g*ph? = 2. _From this solvable equation, we pass to t 2 pu 2 = l by 

 setting (g-\-h^p} 2 = 2t+2u^p. The expressions for g and h in terms of 

 trigonometric functions can be found as before by use of x = i, but are 

 not given. 



113 Jour, fur Math., 17, 1837, 286-290; Werke, I, 343-350. Reproduced by P. Bachmann, 



Die Lehre . . . Kreistheilung, 1872, 294-9. 



114 Dirichlet's Werke, II, 418. 



