382 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xn 



0, 1, S as the initial triple and M, N, P as any triple, derive the next triple by 



A - M 2 

 M^NP-M, 2Vi=--rp, Pi = 



where [k] is the largest integer ^k. When we reach a triple with P = 2S, 

 we get a solution. Application is made to solve nt 2 k= D. 



Several writers 166 proved that, if r is the least integer for which Ar 2 1 = D 

 and if AR 2 +l = D, then R is a multiple of r. 



D. S. Hart 167 showed that 560 is the least z making 953z 2 -f 872+1 = D. 



A. Martin 168 noted that # = 1284836351 gives the least solution of 

 x 2 5658?/ 2 = l, whereas Barlow" gave erroneously a number of 48 digits. 

 [Barlow solved x 2 56587?/ 2 = 1; the omission of 7 was a misprint.] 



H. J. S. Smith 169 proved that, if T, U are the least integral solutions of 

 T 2 DU 2 =( l) i , then T+C/Vl) equals the product of the i complete 

 quotients in a period in the development of V-D as a continued fraction. Also 

 theorems on the number of different periods of complete quotients. 



D. S. Hart 170 gave a " new " method to solve x 2 -Ay 2 = 1. Set A =r 2 m. 

 Then (x-\-ry}(x ry) = l=b?m/ 2 . Set # n/ = l. Then 



But the solutions are not in general integers. He and A. Martin 171 found 

 positive integral solutions of 94x 2 + 57^+34= D. 



A. Kunerth 172 required rational values of p for which x N/D is an 

 integer, N and D being given quadratic functions of p with integral coeffi- 

 cients. Replacing p by a suitable linear function of q, we get* 



dq+f S 



, 

 y y y 



where S=f 2 gd 2 is known. Any common factor of d, f may be removed 

 from each member of the second equation. Write vfw for the rational 

 number q and equate each positive or negative factor of S in turn to 

 v^ gw 2 . Hence for g negative, there is only a finite number of trials. To 

 apply to y 2 = ax 2 +bx+c with the given solution xi, y\, set y = p(x 

 my 2 = a(x 2 x 2 } )+b(xXi)-{-y 2 i. We get 



p 2 a 



The case 6 = is treated at length. The method is applied (pp. 24-32) to 

 Pell's equation y 2 = ax*+l; as y = px+l, x= 2p/(p- a). He reproduced 

 (pp. 56-8) Tenner's rule. 118 



160 Math. Quest. Educ. Times, 23, 1875, 109-110; 24, 1876, 109-111. 

 167 Ibid., 25, 1876, 97. 



108 The Analyst, Des Moines, 2, 1875, 140-2; Math. Quest. Educ. Times, 26, 1876, 87; 

 Math. Magazine, 2, 1890, 59. 



169 Proc. London Math. Soc., 7, 1875-6, 199-208; Collectanea Mathomatica, Milan, 1881, 



117; Coll. Math. Papers, 2, 1894, 148. 



170 Math. Quest. Educ. Times, 28, 1878, 29-30. 



171 Ibid., 101-2; 24, 1876, 39-40. 



172 Sitzungsber. Akad. Wiss. Wien (Math.), 75, II, 1877, 7-58. 



* There are five errors of signs on pp. 15-16. In the examples the signs are correct. 



