CHAP. xii] PELL EQUATION, ax^+bx+c^D. 373 



or II a number equal to the one above it. Then column I gives the de- 

 nominators (quotients) and VI the complete quotients in the continued 

 fraction; if the repeated number (7 in our example) occurs in VI, the last 

 number 2 in I is the last term of the first half of the symmetrical period 

 with an even number of terms;* but if the repeated number occurs in II, 

 the last number in I is the middle term of the symmetrical period with an 

 odd number of terms. If if ax~= 1 is solvable, let L/l, M/m be the 

 last two convergents for Va, the second corresponding to the last quotient 

 in the first half period; it is stated that x = l 2 +m 2 , y = Ll+Mm [cf. Euler, 72 

 end]. For example, if a = 113, cv = 10, the half period is 10, 1, 1, 1, 2 and 

 the convergents are 10/1, 11/1, 21/2, 32/3, 85/8, whence z = 3 2 +8 2 = 73, 

 ?/ = 3-32+8-85. But if we use 1, 1, 1, 2 and the convergents 1/1, 1/2, 2/3, 

 5/8 to Va-a, we have the same x, while y = a(P+m?') + l\+miJL. If there 

 be an odd number of quotients a, - , 2a, let K/k, L/l, M/m be the last 

 three convergents for Va, the third corresponding to the middle quotient; 

 it is stated that x = (k+m)l, y = (K+M}ll = (k+m}L^l [equivalent to 

 Euler's y = lM+kL, since kL Kl= 1]. Tenner continued Degen's table 

 from 1001 to 1020. 



Dirichlet 119 proved that, if D is a complex integer not a square, 

 PDu 2 = l is solvable in complex integers and deduced all solutions. It 

 applies 120 without change to the case of real numbers. The proof rests on 

 the lemma : if a is a given complex irrational number, we can find an infini- 

 tude of pairs of complex integers x, y (y^ty such that N(x-ay}<[N(y*), 

 where N(k+bi} = & 2 +6 2 for k and 6 real. Then, since the modulus of r+s 

 does not exceed the sum of the moduli of r and s, 



Since N(y)^l, the lemma gives 



Hence N(x z a?y 2 } remains less than a fixed limit for an infinitude of pairs 

 of complex integers. Now take a= VZ>. Hence x* Dif takes the same 

 value Z + for an infinitude of pairs x, y, and hence for an infinitude of 

 pairs for which the x's and y's differ by multiples of I: 



x 2 -Dy z = xl-Dijl = l, x=x it y=yi (mod I). 



By multiplication, (xx l -Dyy l } 2 -D(xy l -yx l Y = l\ Since xy l -yx l is di- 

 visible by I, also xx l Dyy l is divisible by I. Hence P Du~ = 1 is solvable 

 in complex integers t,u(u^Q). All solutions are shown to be given without 

 duplication by 



(n = 0, =bl, 2, -), 



+ +++ + + + + 2 + 



119 Jour, fur Math., 24, 1842, 328; Werke, I, 578-588. 



120 Abh. Akad. Wiss. Berlin, 1854, 113; Jour, de math., (2), 2, 1857, 370; Werke, II, 155, 176. 



Cf. Dedekind. 141 



