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



o fc = 2aj_i 1, obtained from a 2 Au 2 = l, y : = 2au, ai = Aw 2 +a 2 = 2a 2 1, 

 . He gave an explicit expression for a k and noted the connection 

 with the formula for cos m$ in terms of sin 4> an d cos <. 

 H. van Aubel 196 proved the statement by Brocard 177 that 



Xm+i = 2px m x m ,i, y m+ i = 2py m y m -i, 



give the relations between three consecutive sets of solutions of x 2 Ay 2 = 1, 

 where p, q give the least solutions. Also theorems giving p and q in terms 

 of the convergents found near the middle of the period of the continued 

 fraction for VZ. If the period has an odd number of terms, A is a sum 

 of two relatively prime squares, but not conversely. He treated values 

 of A, b for which the solution x = by+l, y = 2bf(A-b 2 ) of x 2 -Ay 2 = l is 

 integral. He noted cases when integral solutions can be derived from two 

 sets of fractional solutions. 



Several 197 solved the problem to find the polygons the number x(x 3)/2 

 of whose diagonals is a square, by treating (2v I) 2 8tt 2 = l. 



H. Richaud 198 found the least solution of x 2 -Ny 2 =-l for N = 1549. 

 He noted corrections to Legendre's 88 table for N = 823 and 809. 



J. Vivante 199 treated Dx 2 3 = ?/ 2 (cf. binary quadratic forms). 



E. Lucas 200 gave periods of the continued fraction for Vn, when n is a 

 quadratic function. 



Several 201 solved x 2 -19y 2 = Sl. 



J. Perott 202 reviewed various classic papers on t 2 Du 2 = 1 and proved 

 that, if q is a prime of the form 16n+9, t 2 2qu 2 = 1 is solvable if and only 

 if 2 ( -v /4 = - 1 (mod q) ; while, if q is a prime 16n+ 1, the condition 2^~ 1)/4 = 1 

 (mod q) is necessary, but not sufficient. If q is a prime 8n+ 5, t 2 2q 2 u 2 = 1 

 is always solvable; but, if q is a prime 8w+l, a necessary condition is that, 

 in the decomposition q = c 2 -\-2d 2 , d be divisible by 8. This condition is 

 sufficient if q is of the form 16m+9, but not if q 16ra-f-l. 



F. Tano 203 proved that x 2 Ay 2 = 1 is solvable if A = a l a 2 - - >a n , where n 

 is odd and a i} -, a n are distinct primes =1 (mod 4) and if at most one of 

 Legendre's symbols (a,-/a/) is + 1 for i<j. He gave theorems on the 

 case A = 2ai- -a n . 



J. Knirr 204 gave in detail the Indian 30 cyclic method to solve z 2 cx 2 = l, 

 claiming a simplification. This method is said to be much shorter than 

 that by continued fractions. He tabulated the least solutions for c^!52. 



A. Hurwitz 205 developed any real number x into a continued fraction 

 by use of x = a l/Xi, Xi = ai l/x z , , where a n is chosen so that x n a n 

 lies between -1/2 and +1/2. Minnigerode 159 had shown that the de- 



196 Assoc. frang. av. sc., 14, II, 1885, 135-151. 



197 Mathesis, 6, 1886, 162. 



198 Jour, de Math. E16m., (3), 1, 1887, 181-3. Cf. Whitforil," p. 97. 



199 Zeitschr. Math. Phys., 32, 1887, 287-300. 



200 Jour, de math, spdciales, 1887, 1. 



201 Math. Quest. Educ. Times, 48, 1888, 48. 



202 Jour, fur Math., 102, 1888, 185-223. 



203 Jour, ftir Math., 105, 1889, 160-9. 



204 Die Auflosung der Gleiohung 2* - cz 2 = 1, 18. Jahresberich' Oberrealschule, 1889, 34 pp. 



205 Acta Math., 12, 1889, 367-405. 



