CHAP. II] SOLUTION OF ax+by = c. 49 



J. L. Lagrange 22 used the method of Saunderson 13 and noted that the 

 process is equivalent to the usual one of converting b/a into a continued 

 fraction. He 23 gave a more popular account [results as in Lagrange 17 ]. 



C. F. Gauss 24 employed the notations 

 = O,0] = 0a + l, C=[a,0, y^ = yB+a, [or, 0, 7, 5]= &C+B, 



Apply the g.c.d. process to a and 6 which are relatively prime and positive, 

 with a ^ b; let a = ab + c, b = (3c + d, c = yd + e, , m = pn + 1, 

 so that 



a = O, M, ' ' , 7, 0, 1 6 = [n, M, ' , 7, 0]- 



Take z = [>, , 7, 0], y = [M, -, 7, 0, ] Then ax = by + (- 1)* 

 if fc is the number of the terms a, 0, , p, n. Cf . Euler. 38 



Pilatte 25 solved aix + axi = 6, where fli and a are relatively prime, 

 a > ai, by applying the greatest common divisor process: 



a = a\q\ 02, #1 



Replacing a by its value, we get x = x 2 q&i, where x z = (b 

 must be integral. Thus a 2 x { + a^x 2 = b. Proceeding similarly with the 

 latter equation, we get a 3 x z + a z x 3 = b, , z n _i + a n -ix n = b. Eliminat- 

 ing Xn-ij x n -z, , we get x = d= ab =F ax w , where a is an integer deter- 

 mined by the process. 



P. Nicholson 26 gave a method best explained by his example 



500 - llx llx - r 



y = ^~ = 14 - -^ 5 , r = 10. 



Divide 35z by llx r to get the remainder 2x + 3r. Then divide llx r 

 by 2x + 3r to get the remainder x 16r, in which the coefficient of x is 

 unity. The remainder 20 from the division of 16r = 160 by 35 is the least 

 positive x. But in the example 



200 - 5x 5x-r 



2/ = ir - = 18-- n -, r = 2, 



we reach the remainder x + 2r in which the sign is plus; thus 11 2r = 7 

 is the least x. 



G. Libri 27 gave as the least positive integral solution x of ax + b = cy, 

 where a and c are relatively prime, 



sin 



22 Jour, de l'6cole polyt., cah. 5, 1798, 93-114; Oeuvres, VII, 291-313. 



23 Ibid., cahs. 7, 8, 1812, 174-9, 208-9; Reprint of Legons elem. sur math., Stances des 6coles 



normales, 1794-5; Oeuvres, VII, 184-9, 216-9. 



24 Disq. Arith., 1801, 27; Werke, I, 1863, 20; German transl., Maser, 1889, 12-13. 



26 Annales de Math, (ed., Gergonne), 2, 1811-12, 230-7. Cf. E. Catalan, Nouv. Ann. Math., 

 3, 1844, 97-101. 



26 The Gentleman's Math. Companion, London, 4, No. 22, 1819, 849-60. 



27 Mem. pre"sentes pars divers savants a 1'acad. roy. sc. de 1'Institut de France, 5, 1833, 32-7 



(read 1825); extr. in Annales de Math., ed., Gergonne, 16, 1825-6, 297-307; Jour, fur 

 Math., 9, 1832, 172. Cf. A. Genocchi, Nouv. Corresp. Math., 4, 1878, 319-323. 

 5 



