CHAP. II] ONE LINEAR EQUATION IN n>3 UNKNOWNS. 73 



Thus (1) has the solution a = Xot , = yfa y = z , and also 



x = a cx Q (f> 6bfo, 

 y = /3 - C7/o0 + Oa/8, 

 z = 7 + 60. 



The latter give all integral solutions of (1) when <j> and take all positive 

 and negative integral values and zero. A like result was given by F. 

 Giudice. 156 



*H. Ruoss 157 showed graphically how to find those values of x, y, z in 

 (1) which satisfy certain restrictions, e.g., are all positive. 



ONE LINEAR EQUATION IN n > 3 UNKNOWNS. 



Brahmegupta 2 and Bhascara 5 assigned values to all but two of the 

 unknowns. 



T. Moss 158 tabulated the 412 sets of positive integral solutions of 



170 + 2lx + 27y + 36z = 1000. 



C. F. Gauss 159 noted that, if the constant term is a multiple of the 

 g.c.d. g of the coefficients of the unknowns, then g is a linear function of 

 those coefficients, and the equation is solvable in integers. 



V. Bouniakowsky 147 would solve ax + by + cz + du = by adjoining 

 three equations a'x + = h r , etc., and solving the system. The general 

 solution of the given equation is thus 



x = dp cq + br, z = dr f bq f + aq, 



y = dp' + cq' ar, u = cr' + bp f ap, 



where p, q, r, p f , q f , r' are arbitrary. He gave a like result for five unknowns 



and outlined the law for n unknowns. V. Schawen 160 gave the same method. 



B. Jaufroid 161 assumed that there is no common divisor of a, , m in 



(1) ax -f- by + cz + + rnu + n = 0. 



First, let a and b be relatively prime. Then 



are solvable for A, A i, , so that (1) is satisfied by 



x = Az + Bv+ " + Lu + M - bt, 

 y = A& + Biv + + Lju + MI + at. 



Second, let d be the g.c.d. of a, b and let 5 and c be relatively prime. Set 



a = a : 8, b = 616, and 



(2) aix + biy = p. 



156 Giornale di Mat., 36, 1898, 227. 



157 Korresp. Bl. f. d. hoheren Schulen Wiirttembergs, Stuttgart, 9, 1912, 481-4. 



168 Ladies' Diary, 1774, 35-6, Quest. 658; T. Leybourn's Math. Quest. L. D., 2, 1817, 374-6. 



169 Disquisitiones Arith., 1801, art. 40; Werke, I, 32. 



160 Zeitschr. Math. Naturw. Unterricht, 9, 1878, 111-8. 



161 Nouv. Ann. Math., 11, 1852, 158. 



