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



must be a multiple du of d; the equations Ax By = u, du cz = e are 

 each solved by Fermat's theorem; similarly for n variables (pp. 327-338). 

 A. L. Crelle 63 noted that ax = 1 (mod m) has the solution a* (m} ~ 1 . 

 A. Cauchy 64 obtained independently the result of Bouniakowsky. 61 

 J. P. M. Binet 65 employed Wilson's theorem to solve ax = 1 + py, 

 when p is a prime. We may take < a < p. Then x = (p l)!/a. 

 Whether p is prime or composite, we may also proceed as follows. Divide 

 p by a and call the quotient q and remainder c^; divide p by 0,1 and call the 

 quotient qi and the remainder a 2 ; etc., until the remainder a n = 1 is 

 reached. Then 



aqqi- >q n -i + (- l) n+1 = pM, x = (- l) n qqi- - -q n -i. 



V. Bouniakowsky 66 employed (p,ri) = p(p 1) (p n + 1). Then, 

 if&<p, 



(P + 6, P) = (P, P) + (j ) (P, P ~ !)(&, 1) + (3) (P> P ~ 2) (&, 2) 



Divide by (p, p) and write a = p + b. We get '=! + pJf, where E 

 and 1C are integers* if p is a prime. Hence we have solved ax 1 + py 

 in integers if a > p and p is a prime. To solve 



M x - Ny = 1, N = 

 where p, q, r, are distinct primes, determine i, /?i, so that 



Mai pfii = 1, Ma 2 q(3 2 = 1, Ma 3 r/? 3 = 1, , 

 as above. Raise Mai 1, Jlf2 1, to the powers X, /*, -. Then 



Jlfd + (- l) x = p x A x , Me, + (- 1)" = W, 

 where ei, e 2 , , and vl below are integers. By multiplication, 



- = NB, B = pffa* . 



According as X + M + J' + is odd or even, y = B or B. 



L. Poinsot 67 noted that Lx My = 1 has the solution x = L m ~ l if 

 L m = 1 (mod M), e. g., if m = ^>(M). He also expressed the method in 

 terms of regular polygons. Thus, for 12x 7y = 1, take 7 points PI, , 

 P 7 . Take the first, the fifth after the first, etc. (5 being 12 7) ; we get 

 PiPsPiPiPiPbPz. Since P 2 is now the third point after PI, we have x = 3. 

 We get y from the equation or by use of 12 points. 



63 Abh. Akad. Wiss. Berlin (Math.), 1836, 52. 



"Comptes Rendus Paris 12, 1841, 813; Oeuvres, (1), VI, 113. Exercices d'Analyse et de 



Physique Math., 2, 1841, 1; Oeuvres, (2), XII. See Vol. I, p. 187, of this History. Cf. 



report by J. A. Grunert, Archiv Math. Phys., 3, 1843, 203. 



66 Comptes Rendus Paris, 13, 1841, 210-3. 



68 Mem. Acad. Sc. St. Petersbourg (Math. Phys.), (6), 3, 1844, 287. 



* E = (p + b - 1) ! -=- {p! 6!} is an integer by Catalan, 21 p. 265 of Vol. I of this History. 



67 Jour, de Math., (1), 10, 1845, 55-59. 



