368 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xii 



C. Kramp 97 treated periodic continued fractions and application to 

 Ay-+l = D. The error (p. 283) on ll?/ 2 +49 = x 2 was corrected in the second 

 note. 



P. Tedenat 98 stated that, if y 2 Ax 2 = B is solvable in integers, its 

 solution reduces to the integration of the equation yt+z 2my <+i+2/< = in 

 finite differences, the integral being y = (r+s)/2, x = (r s)/(2VA), where 



Y, X being the least integral solutions of Y 2 AX 2 = B, and m, n being 

 integral solutions of m 2 An? = 1. This is Euler's 72 result in changed nota- 

 tion. 



P. Barlow 99 gave 15 theorems on x 2 Ny 2 = l and the fundamental 

 solution for ./V = 102. He 100 gave general formulas for the solution of 

 x*Ny* = A or z 2 . 



C. F. Degen 101 gave in his introduction an account of y z = ax z +I by 

 the development of Va into a continued fraction, and its solution by an 

 artifice for certain a's, as a = p 2 l, p 2 2. His table I (pp. 3-109) gives, 

 for a ^1000 and not a square, the solutions of y 2 = ax 2 -\-l and the continued 

 fraction for Va [errata, Cunningham 259 - 309 ]. For example, in the entry 



209 [ = a] 14 2 5 3 (2) 

 1 13 5 8 (11) 

 3220 [ = x] 

 46551 [ = ?/] 



the first line gives the continued fraction 



2+5+3+2+3+5+2+28+2+ 



The second line shows auxiliary numbers 1, 13, 5, 8, 11, 8, 5, 13, 1 arising 

 in the process. Thus, 



13 R+12 1 



Table II (pp. 109-112) gives the solutions of y 2 = ax 2 I when solvable 

 [omitted when a is of the form t 2 +l, when y = t, x = l is a solution]. It is 

 said to be solvable only for those values (+2, 5) of a which correspond in 

 table I to a period with an even number of terms. For extensions of 

 Degen's tables, see Bickmore, 219 and Whitford, p. 398 below. 



w Annales de Math, (ed., Gergonne), 1, 1810-11, 261-285, 319-320, 351-2. 

 08 Ibid., p. 349. 



99 Theory of numbers, London, 1811, 294. In x- - 565S7?/ = 1, the figure 7 is omitted; 



cf. A. Martin, Bull. Phil. Soc. Washington, 11, 1888, 592, and Martin. 163 



100 New Mathematical Tables, London, 1814, 266. 



101 Canon Pellianus sive tabula simplicissimam aequationis celebratissimae y z = ax 2 + 1 



solutionem pro singulis numeri dati valoribus ab 1 usque ad 1000 in numeris rationalibus 

 iisdemque integris exhibens, Havniae [Copenhagen], 1817. 



