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



Frattini 210 reduced the solution of x 2 Dy 2 = N to the solution of one 

 of the equations x 2 Dy 2 = Np x , where p = 2m+ 1 n > 0, m 2 being the largest 

 square <D = ra 2 +n. Let x, y be a solution of the given equation such that 

 Then x^(m+l)y. Let x=(m+l)y h, whence h^Q. Then 



our equation becomes a quadratic for y; the radical in the root y must be 

 an integer k. Thus 



79 y.,, 

 k 2 Dh 2 = 



The sign before k must be plus. Hence if y^ ^Njp, and if k, h give positive 

 integral solutions of x 2 Dy 2 = Np, positive integral solutions of x 2 Dy 2 = N 

 are given by 



Applying this result to the new equation, we conclude that, if h= ^N, posi- 

 tive integral solutions of the proposed equation are given by 



k', h' being positive integral solutions of x 2 Dy 2 = Np 2 . The reciprocal of/ 

 is ra+1 VZ><1. Thus we finally reach an equation x 2 Dy 2 = Np* with 

 a solution y exceeding ViVp x /p, and hence a solution of the proposed equa- 

 tion. 



Frattini 211 used similarly x 2 -Dy 2 = N(-ri)*, X = l, 2, , to solve 

 x 2 Dy z = N, and applied the two methods to x 2 Dy 2 = N. He 212 deduced 

 the theorem of Tchebychef. 131 



Frattini 213 supplemented and interpreted geometrically the theorem of 

 Tchebychef. From Frattini 209 we derive the result : If 0, q it q 2 , are 

 values of y in successive positive integral solutions of x 2 Dy 2 = l, the series 

 0, qi^fN, q2^!N, - - separate the positive integral solutions of x 2 Dy 2 = N 

 in such a way that the number of solutions, in which y equals or exceeds 

 any number of that series and is less than the following, is constant. The 

 geometric interpretation is that the vectors of the successive solutions of 

 x 2 Dy 2 = l divide the angle between the positive re-axis and the line of 

 slope I/ VZ> into consecutive angles each of which contains an equal number 

 of points with integral coordinates satisfying x 2 Dy 2 = N. Again, if 1, 

 Pit Pz, are the values of x, the series 0, VAT(pi+l)/2Z>, ^N(p 2 -}-l">/2D, 

 - - separate the solutions of x 2 Dy 2 = N as before; for interpretation, 

 use the ?/-axis instead of the re-axis. 



C. A. Roberts 214 gave only the denominators of the continued fractions 

 for Vp, where p is a prime 4n+ 1^10501 (thus giving what corresponds 

 only to the first line of each entry in the table by Degen, 101 and not the least 

 solution of x 2 py 2 = 1). The introduction to the table is by A. Martin. 



210 Periodico di Mat., 7, 1892, 7-15. 



211 Ibid., 49-54, 88-92, 119-22. 



212 Ibid., 123-124, 172-7. 



213 Atti Reale Accad. Lincei, Rendiconti, (5), 1, 1892, Sem. 1, 51-7; Sem. 2, 85-91. 



214 Math. Magazine, 2, 1892, 105-120. 



