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



Then (1) becomes 8p + cz + + mu + n = and is satisfied by 

 z = B 2 v + . . . + L 2 u + M 2 + 8t, p = B z v + + L 3 u + M 3 ct. 



For these values, (2) becomes a\x + b^y B 3 v - 0. Thus by the 

 first case we obtain solutions x, y, z in terms of v, - - , u, t, t'. A similar 

 method applies when the g.c.d. of a, b, c is prime to d, etc. 



V. A. Lebesgue 162 noted that if a and b are relatively prime, we may set 

 aa + &/? = 1; then the general solution of (1) is 



x = Qa + bw, y = Q(3 aw, Q= n cz mu. 

 But if no two of the coefficients are relatively prime, proceed as for 



4~ + 0,5X5 = 06. 



Set AI = ai, and let A,- be the g.c.d. of a!, -, a, for i = 2, , 5. Remove 

 from 6 the necessary factor A 5 , so that now A 5 = 1. Determine integers 

 di, & such that A;/3; + Oi +1 a t - + i = Ai+i (i = 1, , 4). Solve each of 

 Ai_i?/i_i + 0,-Zf = A;?/* (i = 2, 3, 4), where yi = Xi, and A 4 ?/ 4 + 05^5 = 06- 

 Thus 



2/4 = 



for i = 2, 3, 4. Eliminating the 2/'s, we get x\ t , x 5 in terms of the 

 parameters z 2 , , z 5 . 



E. Betti 154 gave without proof a formula for the integral solutions of 

 + + a n x n = a, where a\, , a n have no common divisor, and 



> n is a particular set of solutions : 



J_ n H I | /j' | /j- 



0103 0203 ~\~ ' ' ' ~\~ 0n 10n-l "T 00n > 



*....* 



x r, ft' rt Q( n ty I r-. /)( n 2) 



- 0i0 n -l -- 020 n -l - - 0n-20_] T UnVn , 



/ /* O\ ^*i O\ 



^ n n H n /)( **""') /3' *' 



X/i C^TI "" Q]yn " ~ d%v n " ~ Cln 2^n " (*n \"n y 



where the n(n l)/2 numbers 0^ are arbitrary. F. Giudice 156 proved that 

 every solution is of this form and gave a method (based upon equations in 

 two variables) of obtaining the general solution in terms of n 1 para- 

 meters. 



C. G. J. Jacobi 163 treated in several ways the solution of 



where/ is the g.c.d. of i, , . Let [0, 6] be the g.c.d. of 0, b. One 

 obtains easily all solutions of 



I / / 



/2?/2 + "3X3 = /32/3, /3 = 



/4 = [/3, 04], 



182 Exercices d'analyse numdrique, 1859, 58. 



188 Jour, fiir Math., 69, 1868, 1-28; Werke, VI, 355-384 (431). 



