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



De Jonquieres 236 noted that a solution of PDu?= 1 or ra 2 can be 

 found from two similar transformations of a quadratic form (A, B, C) into 

 (a, b, c) or its inverse ( a, b, c). 



R. W. D. Christie 237 found the least solution of z 2 -103?/ 2 = l. 



G. Ricalde 238 stated that if x = l, x lt x 2 , - - ; 2/ = 0, y 1} y 2 , are the 

 integral solutions of x z Ay 2 = l (A not a square), 2(z 2 +l) is a square t z 

 and 2/ 2 is a multiple of ; if 2(x 2n+l 1) is a square for one value of n, it is 

 a square /c 2 for every n, and y 2n +i is a multiple of k, and one has the solutions 

 of u 2 Av~ = 1. A. Palmstrom (pp. 210-11) noted that the first statement 

 follows from 



n VI) 2 , 



As to the second statement, Palmstrom proved that (a^n+iTl^iCiTl) are 

 squares, whence 2(x 2n+ i 1) is a square for every n if fbr one n. If Ui, Vi 

 are the least solutions of u z Av 2 = 1, 



so that 2(x 2n +i 1) is a square. But the latter may be true when 



is impossible. 



A. Goulard 239 proved that, if m is odd, 2(x mp 1) is a square if and only 

 if 2(x p l} is a square. The latter is not a square if p is even, while, for 

 p odd, it is a square if and only if u?Av 2 = 1 is solvable. 



A. Cunningham and R. W. D. Christie 240 each noted that Z 2 -p7 2 =l 

 becomes x 2 py 2 =^2 under the transformation Z = z 2 l, Y = xy. Then 

 if p is a prime, it is of the form 8n+3 or Sn 1 according as the upper or 

 lower sign holds. By choosing values of x, y, we get solutions of the 

 proposed equation. 



C. de Polignac 241 proved that if h, HI are the least positive solutions of 

 P Du? = l, where D is positive and not a square, and t n , u n any other 

 solutions, there exists a h'near substitution x i = (QiX-{-S 1 }/(PiX+Ri) whose 

 nth power x n = (QnX+S n )/(P n x+R n ) gives t n = Q n , u n = P n /Ui. 



G. Ricalde 242 gave the identities solving z 2 Ay 2 = l: 



l, (8n+25) 2 -(4n 2 +25n+39)4 2 = l, 



as well as those due to Euler. 65 He and others 243 made minor remarks on 

 the linear relations between three successive solutions of x 2 aif 1. 



236 Comptes Rendus Paris, 127, 1898, 596-601, 694-700. Slightly different from Gauss. 93 



237 Math. Quest. Educ. Times, 70, 1899, 51. 



238 L'interme'diaire des math., 6, 1899, 75. 



239 Ibid. 7 1900 93. 



240 Math. Quest.' Educ. Times, 73, 1900, 115-7. 



241 Ibid., 75, 1901, 67-8. 



242 L'interme'diaire des math., 8, 1901, 256. The third identity lacked the first exponent 2. 



243 Ibid., 59, 286-7. 



