CHAP. XII] PELL EQUATION, ax 2 -\-bx+c=n. 347 



We may deduce roots for additive unity from roots for additive 

 4( 69-72, pp. 365-6). If CL 2 + 4 = G 2 , then L(G 2 - l)/2 and 

 G(G~ 3)/2 are corresponding least and greatest roots for additive unity. 

 If CL 2 - - 4 = G\ and we set p = (G 2 + 1)(G 2 + 3)/2, then pLG and 

 (p l)(Cr 2 + 2) are corresponding least and greatest roots for additive 

 unity. 



If the coefficient C be a square ( 73, p. 366), divide the additive by any 

 assumed number 6. To the quotient add b and from it subtract b and di- 

 vide by 2. The first result is a greatest root; the second, divided by the 

 square root of (7, is the corresponding least root. 



If the coefficient be divisible by a square t 2 (75, p. 367), use the quo- 

 tient as a new coefficient and find roots. If the least root so found is divided 

 by t, we get the desired least root. The greatest root rema ns the same. 



For C = 3, A = - 800 ( 77, p. 368), remove the factor 20 2 . For the 

 new additive 2, we get roots 1 and 1. Their products by 20 are the roots 

 desired. 



Alkarkhi 29 (about 1010) solved x 2 + 5 = if by setting y = x + 1, and 

 x 2 - 10 = y 2 by setting y = x - 1. To solve 77 2 z 2 - 160 = w 2 , set 

 w = 77x - 2. To solve (pp. 72-4) x 2 + 4z = y 2 , setjy = 2z; to solve 

 4x 2 + IQx + 9 = y 2 , set y = 2x n, where n 2 > 9, say n = 5. As the 

 condition (p. 113) for rational solutions of (ax 6) x 2 = D, he found 

 that Ja 2 T b must be a sum of two squares. Finally (p. 121), v 2 w 2 = a/3 

 for v = (a + |8)/2, w = (a - 0)/2. 



Alkarkhi 29a used the approximation a + r/(2a + 1) for Va 2 + r. 



Ibn Albanna 296 (born about 1255) used the same approximation when 

 r > a, but for r ^ a employed a + r/(2a). The latter was used by Heron 

 of Alexandria and by Elia Misrachi (1455-1526) in his Arithmetic (ed., 

 G. Wertheim, 1893, 1896). 



Bhascara Acharya 30 (born 1114) gave a method of deducing new sets of 

 solutions of Cx 2 + 1 = y 1 from one set found by trial. Take any number 

 4= and call it the "least root" L [for additive A]. By the positive or 

 negative additive quantity A is meant a number which added to or sub- 

 tracted from CL 2 makes the sum or difference a perfect square, its root being 

 called the "greatest root" G. Thus if (7 = 8, L = 1, A = 1, then G = 3. 



Composition ( 76-77, p. 171). From these roots L, G and the same 

 or a new set of roots I, g, we obtain by cross multiplication and addition a 

 new least root X = Lg + IG, while 7 = CLl + Gg is the corresponding new 

 greatest root. The product of the two additives gives the new addit ve. 

 Thus (82) for the former example, take 1 = 1, g = 3, A = 1;' then 

 X = 6, 7 = 17. Next, from L = 1, G = 3 and X = 6, 7 = 17, we get the 

 new roots 35, 99 and so on indefinitely by means of composition. 



"Extrait du Fakhri, Traite d'algebre par Ben Alhagan Alkarkhi (Arab MS.), French transl. 



by F. Woepcke, Paris, 1853, 84, 120. 

 290 Kafi fil Hisab, German transl. by A. Hochheim, II, 14. 

 296 Le Talkhys, p. 23. French transl. by A. Marre, Atti Accad. Pont. Nuovi Lincei, 17, 



1864, 311. 

 30 Vija-gan'ita (algebra), Ch. 3, 75-99, "Affected square." Colebrooke, 28 170-184. 



