48 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



Lagrange 18 proved as had Euler 14 that, if b and c are relatively prime, 

 there exist integers y and z such that by cz = a. Next, if y = p, z = q 

 is one set of solutions, every set of solutions is given by y = p + we, 

 2 = q + nib. If a = a'd, c = c'd, where a' and c' are relatively prune, then 

 p is divisible by d, say p = p'd. As in the proof of the initial theorem, we 

 can find m such that p' + me' is divisible by a'. Hence we can always 

 find a value of y which is a multiple ar of a; then z is a multiple a's of a', 

 and br c's = 1. From a set of solutions r, s of the latter, we get 

 y = ra + we, = sa' + nib. 



Lagrange 19 noted that his 17 method is " essentially the same as Bachet's, 6 

 as are also all methods proposed by other mathematicians." To solve 

 39# 56y = 11, employ 



56 = 1-39+17, 39 = 2-17+5, 17 = 3-5+2, 5 = 2-2+1, 2 = 2-1. 

 By means of the quotients 1, 2, 3, 2, 2, we get the convergents 



1 3 10 23 56 



1' 2' 7' 16' 39' 



Thusz = 23-11 + 56m, y = 16-11 + 39m. 



L. Euler 20 employed the method of always dividing by the smaller 

 coefficient, thus following Rolle 8 in essence. For 5x = 7y + 3, 



The numerator must be a multiple of 5. Thus 2y + 3 = 5z, 



whence y = 5u + 6, x = 7u + 9. He showed that the process is equivalent 

 to that for finding the greatest common divisor of 5 and 7 : 



7 = 1-5 + 2, x = l-y + z, 



5 = 2-2 + 1, y = 2-2 + u, 

 2 = 2-1 + 0, z = 2-u + 3. 



Jean Bernoulli 21 applied Lagrange's 19 method to find the least integer 

 u giving an integral solution of A = Bt Cu, when B, C are relatively 

 prime, in the special cases A = |C, |C + 1, %(C 1). For example, if C 

 is even and A = C/2, then u = (B - - l)/2, t = (7/2. If C is odd and 

 A = |(C + 1), then M = |( + s - 1), t = %(C + r), where Br - Cs = 1, 

 r/s being the convergent just preceding C/B in the continued fraction for 

 the latter. 



18 Mem. Acad. Berlin, 24, annexe 1768, 1770, 184-7; Oeuvres, II, 659. 



19 Ibid., 220-3; Oeuvres, II, 696-9. Additions by Lagrange to Vol. 2 of the transl. by Jean 



III Bernoulli of Euler's Algebra, Lyon, 1774, 517-523 (Euler's Opera Omnia, (1), 1, 1911, 

 574-7; Oeuvres de Lagrange, VII, 89-95). 



20 Algebra, 2, 1770, 4-23; French transl., Lyon, 2, 1774, pp. 5-29; Opera Omnia, (1), I, 



326-339. 



21 Nouv. Me"m. Acad. Roy. Berlin, ann6e 1772, 1774, 283-5. 



