76 HISTORY OF THE THEORY OF NUMBERS. [CHAP. II 



R. Ayza 168 treated ax + by + cz + du + = k by means of 



ax + fa/ = ki, cz + du = k z , - , ki = k k z k 3 , 



where & 2 , k 3 , are arbitrary. For m linear equations in m + n variables, 



successive elimination gives one equation in m + n variables, one in 



m + n 1 variables, , one in n -\- 1 variables, which is solved as above. 



A. P. Ochitowitsch 169 treated 2ai?/i = 0. If a p , a q are relatively prime, 



y p = za q + ry' p , y q = - za p + r^, a p y' p + a a ^ + 1 = 0, 



where r = 2a t -i/i for i ={= p, g. For i = p 1 - -p n , where pi, , p n are 

 distinct primes, a set of solutions of 1 + a-[X\ + 2 ^2 = is given by 



where Zi, , z n are to be chosen to make the indicated fractions integral. 

 E. B. Elliott 170 recalled the fact that all sets of positive integral solutions 

 of a linear diophantine equation in n variables are linear combinations of 

 a finite number of " simple " [fundamental^ sets of solutions (i, , ), 

 , (coi, , w n ) and that a linear combination of these simple sets is 

 always a solution. He noted that two such combinations may give the 

 same solution since the simple sets are usually connected by syzygies. 

 For example, the three simple sets (103), (230), (111) of solutions of 

 3x = 2y + z are connected by the syzygy (103) + (230) = 3(111), so that 



x = t l + 2t 2 + t 3 , y = 0i + 3t 2 + t 3 , z = 3*! + Ok + U 



give duplicate solutions unless we restrict U to the values 0, 1, 2. In this 

 sense, he obtained formulas giving each solution of an equation in n variables 

 once and but once, making use of generating functions. 



G. Bonfantini 171 noted that, if a, , I, k have no common factor, 

 ax + by + + lu = k has integral solutions if and only if a, , I 

 have no common factor. 



Several 172 found all positive integral solutions of 



13k + 211 + 29m + 37n = 300. 



* P. B. Villagrasa 173 treated (3). 



D. N. Lehmer 173a proved that (3) with A = 1 is satisfied by the co- 

 factors of the elements of the last row of a certain determinant of value 

 unity, those elements being any integers whose g.c.d. is 1. The general 

 solution of (3) is deduced. 



188 Archive de Matematicas, Madrid, 2, 1897, 21-25. 



189 Text on linear equations, Kasan, 1900. 



170 Quar. Jour. Math., 34, 1903, 348-377. 



171 II Boll, di Matematica, Gior. Sc. Didattico, 3, 1904, 45-47. 



172 Math. Quest. Educ. Times, 7, 1905, 21-22. 



173 Revista de la Sociedad Mat. Espanola, 3, 1914, 149-156. 



1730 Proc. Nat. Acad. Sc., 5, 1919, 111-4; Amer. Math. Monthly, 26, 1919, 365-6. 



