CHAP. II] SYSTEM OF LINEAR EQUATIONS. 77 



Since the problem to solve n = ax + by + in positive integers is 

 the same as to partition n into parts a, b, , reference should be made to 

 Ch. Ill, in particular for theorems on the number of solutions. 



SYSTEM OF LINEAR EQUATIONS. 



Chang Ch'iu-chien 174 (sixth century A.D.) treated a problem equivalent 

 to 



x + y + z = 100, 5x + 3y + fz = 100, 



and gave the answers (4, 18, 78), (8, 11, 81), (12, 4, 84). 



Mahaviracarya 175 (about 850 A.D.) treated special cases of 



x + y + z + w = n, ax + by + cz + dw = p. 



Shodja B. Aslam 176 (about 900 A.D.), an Arab known as Abu Kamil, 

 found positive integral solutions of x + y + z = 100, 5x + ?//20 + z = 100, 

 whence y = 4x + 4z/19, x = 19; x + y + z = 100 = fa; + \y + 2z, 

 whence x = 60 9?//10, y = 10m, m = 1, , 6; 



X |~ y ~T" Z ~\ U JLUU, QX "T" \~y "T" 2^ ~T~ W 1UU, 



whence a: = ^i/ + |z, with 98 sets of solutions (two of which are omitted). 

 When the last equation is changed to 2x + \y + \z + u = 100, there are 

 304 sets of solutions. There is no solution of 



x + y + z = 100 = 3z + 2//20 + z. 

 There are 2676 sets of positive integral solutions of 



x + y + z + u + v = 100, 2x + \y + \z + \u + v = 100. 

 Alhacan Alkarkhi 177 (eleventh or twelfth century) treated the system 

 \x + w = \s, f y + w = \s, |z + w = Js, 



s = x + y + z, ws H2"^"3"^6/ > 



by taking s = l, whence # = 33, y = 13. He treated the problems of 

 Diophantus I, 24-28, as had Diophantus, by making the indeterminate 

 problems determinate by assigning a value to one unknown. 



Leonardo Pisano, 178 in 1228, treated various linear systems, the first 

 being that of Alkarkhi 177 without the final condition : 



t x t 2y t 5z 



174 Suan-ching (Arith.). Cf. Mikami, 71 43-44. 



175 Ganita-Sara-Sangraha. 3 Cf. D. E. Smith, Bibliotheca Math., (3), 9, 1909, 106-10. 



176 H. Suter, Bibliotheca Math., (3), 11, 1911, 110-20, gave a German transl. of a MS. copy 



of about 1211-8 A.D. 



177 Extrait du Fakhri, French transl. by F. Woepcke, Paris, 1853, 90, 95-100. 



178 Scritti di L. Pisano, 2, 1862, 234-6. Cf. A. Genocchi, Annali di Sc. Mat. e Fis., 6, 1855, 



169; O. Terquem, ibid., 7, 1856, 119-36; Nouv. Ann. Math., Bull. Bibl. Hist., 14, 1855, 

 173-9;' 15, 1856, 1-11, 42-71. 



