402 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xin 



Let Qi be the denominator of its ith convergent. Then 



Q 2n = (2uQ n -i - aQn-2) 2 - aQ*_i, 2uQ n -i - aQ- 2 = Q n . 



Hence a solution is y = Q n , x = Q n -\, bz = Q 2n , the last being used to determine 

 u and n : 



If 6 is odd, set k = (2p 1)6; then (*) is divisible by 6 if p<k, and we may 

 take 2n = k 1. If 6 is even, divide x and y by a power of 2. 



P. Mansion 10 gave a short proof of the preceding QlaQl_ l = Q^ n . 



S. Re'alis 11 noted that, if a, /3, 7 is one solution of ax 2 +bxy-{-cy 2 = hz, a 

 second is given by x=(h+a c)a+(6+2c)/3, 2/=(2a+6)a+(/i 

 since 



= a/i+2a(a+6-c)+6 2 , 



If, for c 1, we solve the initial equation for ?/, the radical will be a rational 

 number u if u 2 Dx 2 = 4hz, D = b 2 4a, which was treated (i6idL, p. Ill) 

 and if D>0 by Giinther. 9 



A. H. Holmes 12 proved that 96x 96i/+21 = D is impossible in integers. 



On ax 2 +bx+c = K see Desmarest. 87 



SOLUTION OF x 2 y 2 = g. 



Diophantus, II, 11, took gr = 60, x = y+3, 3 being a number ^ V60, 

 whence y 17/2. 



Leonardo Pisano 13 took a square a 2 <# and set (z+a) 2 = x 2 +<7, which 

 determines x. He gave a second method. Let g be odd, gr = 2n+l. Since 

 1+3H ----- h(2n l)=?i 2 , we may take y = n, whence n 2 +gr = (n+l) 2 . He 

 treated separately the cases g = 2k, g 4k. 



R. Descartes 14 noted that 6 2 -3 2 = 3 3 , 118 2 -10 2 = 24 3 ; (ax) 2 -x 2 = x 3 if 

 x = a 2 1. 



J. L. Lagrange 15 concluded from his general theory of binary quadratic 

 forms / that every integer is of the form y z z~. This 16 is not true of the 

 double of an odd prime, and Lagrange's argument is conclusive only when 

 the discriminant of / is not a square. 



S. Canterzani 17 treated 2 +A = D, by deciding whether or not A is a 

 sum of differences of consecutive squares. First, let A be even. The sum 

 of 2/ consecutive differences 2/i+l, 2/1+3, is 4//i-f4/ 2 and hence 



10 Jour, de Math., (3), 2, 1876, 341. 



u Nouv. Corresp. Math., 6, 1880, 348-350. 



12 Amer. Math. Monthly, 18, 1911, 70. 



14 La Practica Gcometriae, 1220. Scritti di L. Pisano, Rome, 2, 1862, 216-8. 



" Oeuvres, X, 302, posthumous MS. Cf. papers 23-26 of Ch. XX. 



16 Nouv. m<Sm. Acad. Sc. Berlin, annde 1773; Oeuvres, III, 714. 

 18 L'interme'diaire des math., 18, 1911, 33. 



17 Memorie dell'Istituto Nazionale Italiano, Classe di Fis. e Mat., Bologna, 2, II, 1810, 445-76. 



