CHAP. XII] PELL EQUATION, a 2 +foc+c= D. 387 



velopment is periodic if X Q is a root of a quadratic equation with integral 

 coefficients. The necessary and sufficient condition for the solvability 

 of x 2 -Dy*=-l is that 



\D=(a Q ; i, a z , , a r , a b a 2 , , a r ; a i} a 2 , ) 



G. Chrystal 206 gave an exposition of x 2 Cy 2 = H convenient for English 

 readers. 



F. Tano 207 proved by developing Va 2 4 into a continued fraction that 

 x 2 (a 2 +4)2/ 2 = 1 is solvable in integers when a is any odd integer, while 

 re 2 (a 2 4)?/ 2 = 1 is impossible except when a = 3. There are infinitely 

 many integral solutions of x 2 kz~ = a if a is any odd integer and k a sum 

 of two squares. To prove that there are infinitely many integral solutions of 



where N is any integer, we add the two equations 



x 2 (a 2 -\-^)y 2 = a, x\ (a 2 tyy\= (2a 5) 



if N is odd; but, if N is even, we first change the second members to a, 4. 

 By multiplying x 2 a 2 y 2 4y 2 =a by u* a 2 t>;+4v; = l for i=l, 2, , 

 in turn, we find that there is an infinitude of integral solutions of 



Y 2_y 2- / 3"-l\ 



r=l r-l \ 2 / 



G. Frattini 208 noted that, if or , ?/o is the fundamental solution of 

 x 2 (a 2 +l)?/ 2 = N, viz., a solution with 0<2/ = VA7, then all its solutions 

 are given by 



where n ranges over the values 0, 2, 4, ; while all solutions of 

 x 2 (a?-\-l)y 2 = -{-N are given by the same formula where n ranges over the 

 positive odd integers. 



Frattini 209 proved that, if K, H (H<^N) form a solution of 

 x~ (a 2 l)y 2 = N, every solution in positive integers is given by 



tf-l) m , w = 0, 1, 2, 

 Let a 2 -D|8 2 =l, 0=1=0. Multiplying x 2 -Dy~ = N by 2 , we get 



whose solutions are derived from one by the preceding formula, viz., 

 x+y^D = (K+HJD}(a+t3^D : ) m , m = 0, 1, 2, . 



When N is changed to N, the same formulas hold if we replace K by 

 where, in the final formula, H ^ V#(a+l)/2Z>. Tchebychef 's 131 first result 

 is a corollary. 



206 Algebra, 2, 1889, 450-60; ed. 2, 2, 1900, 478-86. 



207 Bull, des Sc. Math., (2), 14, I, 1890, 215-8. 



208 Periodico di Mat., 6, 1891, 85-90. 



209 Ibid., 169-180. 



