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



values of x + y when a b = 1, 2, , 100. The paper ends with a 

 twenty page bibliography and history of linear diophantine equations. 



C. A. Laisant 42 constructed the points having as abscissas 1, 2, , p 

 and as ordinates the corresponding residues < p modulo p of r, 2r, , pr 

 (r prime to p). The lattice defined by these points leads to an immediate 

 solution of rx pz = a since every point of the lattice has the coordinates 

 x, y = rx pz. 



W. F. Schiiler 43 gave a collection of 374 problems on linear Diophantine 

 equations and an extract from Bachet 6 with a German translation. 



E. W. Davis 44 used points with integral coordinates to solve ay bx k. 



P. Bachmann 45 gave an extended account of Euclid's g.c.d. algorithm, 

 continued fractions, and related questions. 



A. Pleskot 46 treated I3x + 23y = c somewhat as had Rolle 8 : 



c = 13(x + 2y) - 3y = 3(4z + 7y) + x + 2y, 

 4z + 7y = t, x + 2y = c - 3t, x = - 7c + 23t, y = 4c - 13t. 



J. Kraus 47 solved ax a'y = c, where a' a k exceeds a and c, 

 by use of or A r A+1 = ka^ < r A < k, ^ A < a. \ 1, 2, -, thus 

 representing rjk as a number with the digits a x , a A+i , to the base a. 



P. A. MacMahon 48 proved that, if the continued fraction for X//* has a 

 reciprocal series 0,1, a z , , 0,2, i of partial quotients, 2i 1 in number, 

 then the fundamental (ground) solutions of \x = ^y + z are (jc/, y 3 -, y ,+!_/), 

 j = I, , <r, if X > /*, where cr = 1 + i + a 3 + a 5 + + s + s + aij 

 but are (x y , ?/,-, x,_y) and (a;,, 2/ ff , 0), y = 1, , <r - 1, if X < /*, where 

 <r = 1 + a 2 + 4 + + 4 + (h + 1, not including a twice. When the 

 partial quotients are even in number, the fundamental solutions depend 

 upon both the ascending and descending sets of intermediate convergents 

 to X/ju. He 48a had proved that the fundamental solutions are always 

 fa, yj, 2y), j = 1, , a, where the yj/Xj are the ascending intermediate con- 

 vergents tO X/JLt. 



A. Aubry 49 plotted the points with integral coordinates = x < n, 

 = y < n, as well as the lines y = x, y = ax, y = bx, , where 1, a, b, 

 are the integers < n and prime to n. Thus we can read off the integer 

 = yfx (mod n) and hence solve ax nz = g. 



N. P. Bertelsen 490 solved bx cy =z, 1 ^y <b, O^z^c, 1 ^z < &, 

 by use of the convergents b r fc r to the continued fraction (a , i, , fl n ) 

 for b/c. Then ?/ is a linear function with positive integral coefficients of 

 b r + kbr+i (k = 1, 2, -, o r+2 1), and x is the same function of the 



C r + 



42 Assoc. fran<j. av. sc., 16, II, 1887, 218-235. 



43 Lehrbuch der unbestimmten Gl. 1 Grades, Stuttgart, 1, 1891, 176 pp. (Kleyers Encykl.). 



44 Amer. Jour. Math., 15, 1893, 84. 



46 Niedere Zahlentheorie, 1, 1902, 99-153. 



46 Zeit. Math. Naturw. Unterricht, 36, 1905, 403 [33, 1902, 47]. 



47 Archiv Math. Phys., 9, 1905, 204. 



48 Quar. Jour. Math., 36, 1905, 80-93. 



480 Trans. Cambridge Phil. Soc., 19, 1901, I. 



49 L'enseignement math., 13, 1911, 187-203. Cf. G. Arnoux, Arith. Graphique, 1894, 1906. 

 490 Nyt Tidsskrift for Mat., B, 24, 1913, 33-53. 



