CHAP. II] SOLUTION OF ax=b (mod m). 55 



is any odd number relatively prime to* <f>(m), then 



[ab~\ 



( mod 

 Lm J 



1 12 



~ = a ~ ~^T^ 

 a P(m) 



where the summation extends over all positive integers b which are < m 

 and prime to m, while P(m) = (1 p)(l p') , if p, p r , - - are the 

 distinct prune factors of m. 



E. Busche 56 obtained graphically the number of solutions of az = b 

 (mod rn), including solutions called improper or transfinite, 57 introduced 

 when a and m have a common factor > 1. As the ordinary (proper) solu- 

 tions may be restricted to the integers 0, 1, , m 1, we are at liberty 

 to designate the improper solutions by numbers is m. The simplest case 

 is one like 3z = b (mod 15), in which 3 and 15/3 are relatively prime; then 

 there is defined an improper solution designated by 15 if b = 0, 15 + j if 

 b = j (j = 1, , 4), 15 if b = 5, 15 + j if b = 5 +; (j = 1, , 4), etc. 



SOLUTION OF ax = b (MOD ra) BY FERMAT'S OR WILSON'S THEOREM. 



J. P. M. Binet 58 noted that, if a is a prune not dividing 6, bx ay = 1 

 has the solution x = b a ~ 2 , the corresponding y being integral; while, if 

 p, p f , are the equal or distinct prime factors of a, 



bx = 1 - (1 - ftp-'Xl - be'- 1 )-.- 



gives an integer x, leading to an integer y, such that x, y satisfy the same 

 equation. The same method was found independently by G. Libri. 59 

 A. Cauchy 60 expressed Binet's method in the following form: let 



n = a a b -, (1 - k a ~ l } a (l - k^Y - = 1 - kK. 



Then for k prime to n, 1 kK is divisible by n, so that kx = h (mod ri) 

 has the solution x = hK (mod n). 



V. Bouniakowsky 61 proved that, if a, b are relatively prune positive 

 integers, ax T by = c has the integral solutions* 



l t y = 



G. de Paoli 62 gave the last solution, with <(&) replaced by 0(6)/2 when 6 

 is divisible by 4. To solve ax by cz = e, where a, 6, c, e have no com- 

 mon divisor, let a = dA, b = dB, where d is the g.c.d. of a, b; then e + cz 



*By <f>(m) is meant the number of integers < m which are prime to m. 



66 Mitt. Math. GeseU. Hamburg, 4, 1908, 355-380. 



67 Imaginary by Gauss, Disq. Arith., Art. 31; G. Arnoux, Arithme'tique graphique, 2, 1906, 



20. Both excluded such solutions. 



68 Jour, de I'e'cole polyt., cah. 20, 1831, 292 [read 1827]; communicated to the Socie'te' 



Philomatique before 1827. 



69 Mdmoires de Math, et de Phys., Florence, 1829, 65-7. Cf. Libri 148 of Ch. XXIII. 



60 Exercices de Math., 1829, 231- ; Oeuvres, (2), IX, 296. 



61 Me"m. Acad. Sc. St. P<tersbourg (Math. Phys.), (6), 1, 1831, 143-4 [read Apr. 1, 1829]. 



62 Opuscoli Mat. e Fis. di Diversi Autori, Milano, 1, 1832, 269. He stated that the paper was 



written in 1830 without knowledge of that by Binet. 



