CHAP, xii] PELL EQUATION, ax 2 +bx+c=n. 381 



If x=f, y = g give the least solution of ax-y z = 1, all solutions are given 

 by the preceding and a similar formula. 



D. S. Hart 158 stated that, if the fundamental set of solutions p , q of 

 p 2 Nq- = 1 has been found, so that we have a set in addition to 1, 0, the 

 simplest method to find successively all further sets of solutions is to use 

 the relations p = 2p r-\-r', g = 2p s = Fs', where r, s are the last found values 

 of p, q, and r', s' the next preceding values. 



B. Minnigerode 159 modified the theory as presented by Dirichlet 133 by 

 using a different definition of reduced forms and using the continued fraction 



with negative quotients (see the chapter on binary quadratic forms in 

 Vol. III). 



W. Schmidt 160 showed that all positive solutions of t 2 Du?=4 are 

 given by the development into a continued fraction of a root of a certain 

 reduced binary quadratic form of determinant D. 



T. Muir 160a treated the development into a continued fraction of the 

 square root of any positive integer or fraction. In particular, he obtained 

 (p. 19) in general form the results of Euler, 72 calling Euler's (a, , I) a 

 continuant K(a, , I). 



D. S. Hart and W. J. C. Miller 161 proved by use of p 2 -103n 2 = l that 

 103(3x -2) 2 +l = D has no integral solution x and that 22421/3 is the least 

 positive solution. 



M. Collins and A. M. Nash 162 proved that x*+D m = (N 2 +D)y^ is solvable 

 in rational numbers if m = 2n-\-l by taking 



x+Ny = kD(y+D n ), x-Ny=(y-D n )/k. 



Several 163 solved x 2 953i/ 2 = =tl by the continued fraction for V953. 

 S. Tebay 164 noted that, if p, q are the least solutions of x z ny* = l, then 



Let na?k = m?. To solve nt 2 k= D, set t = a+r. Then 

 ra 2 -\-2nar +UT Z = D = (m+rxfyY, if T = 2y(nay 



S. Bills 165 illustrated a "new, practical" method of solving x' 2 Ay 2 =l 

 by taking A = 953. Then $ = 30 is the root of the square just <A. From 



8 Math. Quest. Educ. Times, 20, 1874, 64. 



159 Gottingen Nachrichten, 1873, 619-652. Cf. A. Hurwitz. 205 



160 Zeitschrift Math. Phys., 19, 1874, 92-94. 



icoa The Expression of a Quadratic Surd as a Continued Fraction, Glascow, 1874, 32 pp. 



Cf. R. E. Moritz, Ueber Continuanten und gewisse ihrer Anwendungen im Zahlen- 



theoretischen Gebiete, Diss. Strassburg, Gottingen, 1902. 

 i Math. Quest. Educ. Times, 20, 1874, 66-7; 28, 1878, 65-66. 

 * Ibid., 22, 1875, 23-24. 

 "/&*., 78-80; 23, 1875, 107. 

 "4 Ibid., 23, 1875, 30. 

 "* Ibid., 98-99. 



