CHAP. Xlll] az 2 +fy/ 2 +cz 2 = 0. 419 



U. Fornari 102 treated (x-l)(x-2}+y(2x+y-l')=2m. 



W. A. Wijthoff 103 solved (x+y+l) 2 = 9xy. 



M. Rignaux 103a stated a complete solution of (11), with ac<6 2 , by recur- 

 ring series. 



For $x(x+l) = ?y(y-\-l) see paper 55. For 3x(x+l)=y(y+l), see 

 Euler 79 of Ch. I. T. L. Pistor 107 of Ch. XII gave Gauss' 85 method. On 

 ax 2 -a'x=:by' i -Vy, see Gill 107 of Ch. I. 



ax z +by--\-cz* = Q (EXCEPT 



For # 2 +2/ 2 = 2z 2 , here excluded, see squares in arithmetical progression 

 (Ch. XIV). 



Diophantus, II, 20, proposed to find three squares such that 



(1) y 2 -x* :z*-y z = a :b, 



where a : b is a given ratio. He took a/b = 1/3, y=x+l, whence 



Take z=z+3, whence a; = 5/2. In IV, 45, he took 0/6 = 3, z 2 = 4, y = t+2, 

 whence fz 2 = 3* 2 +12+9 = (3-50 2 , if = 21/11. 



Alkarkhi 104 (beginning of eleventh century) solved x z y 2 = 2(y 2 z-) by 

 taking y = z+l, x=z+2, whence 2 = 1/2. 



Leonardo Pisano 105 first treated (1) for several special cases. For 

 6 = a+l, take x = 2a 1, y = 2a+l, z = 2a+3; then y 2 x- = 8a, z*y z = 8b. 

 In general, if integers h, k, n can be found such that 



then y z -x z = Ska, z 2 -if = 8kb for 



z = 2/H-l, y = 2h+2a+l, z = 2h+2a+2n+l. 

 The conditions for the above sums are 



or 



a+l_n(n+a) 



2 ~2(b-n)' 



These fractions must equal integers, as in the case for the values a = 11, 

 6 = 43, n = 16, k = S, h = 2, used by Leonardo. A. Genocchi 106 remarked 

 that Leonardo's method consists essentially in separating a progression 

 /i+l, h-\-2, -, /i+m+w into two parts such that the sum of the first m 

 terms is ka and the sum of the last n terms is kb, whence 



mn(m+ri) 2amn+an 2 bm? 



2k = 



, 

 bm an urn an 



102 II Pitagora, 19, 1913, 57-60. 



103 Wiskundige Opgaven, 11, 1912-4, 192-5. 

 1030 L'interm6diaire des math., 26, 1919, 9. 



104 Extrait du Fakhri, French transl. by F. Woepcke, 1853, 116. 



5 Tre Scritti, 103-112. Scritti, 2, 1862, 275-9 (Opuacoli). Cf. Ch. XVI. 



106 Annali di Sc. Mat. e Fis., 6, 1855, 351-2 (misprint of sign before bm 2 in the fraction for 



