394 HISTORY OF THE THEORY OF NUMBERS. [CHAP. XII 



is of Euler's form P 2 +QR. (ii) C+Z>=D. (iii) -AD=H. (iv) 

 -CD=H. (v) One of A(l-D), C(l-D), D(l-A), D(l-C) a square, 

 and the generalizations to D = m z Di, A(lDi) = D = <? 2 (etc.), since then 



R. W. D. Christie 252 noted that, if ad-bc = l, a 2 +6 2 = P, x 2 +l=Py 

 is satisfied by 



The problem is now to choose a, b, c, d to make y = D . He and others 

 (p. 87) solved x 2 149?/ 2 = 1 without using continued fractions. He and 

 E. B. Escott (p. 119) gave the identity 



Christie 253 noted that if p n fq n is a convergent to YD, where D is a prime 

 4m+l, then g 2 n+i = Q'+ff+i [cf. Euler, 72 end]. 



G. Frattini, 254 employing a positive integer D and positive rational 

 numbers E, F, defined the index of E+F^D to be the maximum number 

 of such factors into which it can be decomposed. If one solution a, 13 of 

 a; 2 D?/ 2 = l is known, all solutions of x~Dy 2 = N are given by 



where the index of the particular solution x r , y' does not exceed half the 

 index of the solution of the Pell equation. But we may regard as known 

 the solutions whose indices do not exceed a given limit (depending only on 

 a finite number of trials) . 



Frattini 255 extended the preceding results to the algebraic case in which 

 D, N, x, y are polynomials in a parameter a. Finally, he proved that, if 

 D is a positive integer or a polynomial of even degree in a, x 2 Dy 2 = l is 

 solvable if and only if VZ> is developable into a simple periodic continued 

 fraction such that 



a z , --, a n , c+ VS), 



where the o's and c are integers if D is integral, otherwise polynomials in a. 



A. Cunningham 256 gave the least solutions of both r 2 Dir = 1, D< 100, 

 from Degen's 101 table, but checked by Legendre's 88 ; also further (multiple) 

 solutions for D^ 20; also the least odd solutions of r 2 Dir = zb2, 8, 16 

 for D<500, and D=d=4 for ZX1000 (computed from data in Degen's 

 table). He noted three errors in the table by Bickmore. 219 



Cunningham and Christie 257 showed how to find an infinitude of integers 



262 Math. Quest. Educ. Times, (2), 6, 1904, 98-101. 



263 Educ. Times, 57, 1904, 41. 



254 Periodico di Mat., 19, 1904, 1-15. 



265 Ibid., 57-73. Cf. Frattini, 283 H. E. Heine, Jour, fiir Math., 48, 1854, 256-8. 



266 Quadratic Partitions, 1904, 260-6. 



267 Math. Quest. Educ. Times, (2), 7, 1905, 79-80. 



