54 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. II 



SOLUTION OF ax = b (MOD m) WITHOUT FERMAT'S THEOREM. 



C. F. Gauss 50 noted that ax = b (mod m) is solvable if and only if b is 

 divisible by the g.c.d. d of a = de and m = df. Let b = dk. Then 2 is a 

 root of the proposed congruence if and only if ex = k (mod/), while the 

 latter has a unique root modulo /. For a compoiste modulus mn, a second 

 method is often preferable. First, employ the modulus m as above and 

 let x = v (mod m/d), where d is the g.c.d. of m and a. Then x = v + x'm/d 

 is a root of ax = b (mod mn) if and only if x'a/d = (b av)/m (mod n). 



P. L. Tchebychef 51 proved that, if the g.c.d. d of a and p divides b, 

 ax = b (mod p) has the d roots a, a + p/d, , + p(d l)/d, where 

 aa/d = 6/d (mod p/d) . 



C. Sardi 52 considered the congruence a& = b (mod p) in which p is a 

 prime not dividing a\, and b < p. Dividing p by ai, let a 2 be the remainder 

 and [p/aj the quotient, where [n~] is the greatest integer ^ w. Multiply 

 our congruence by [p/ai]', we get 



a 2 x = b [p/i] (mod p). 



Let a 3 be the remainder when p is divided by a 2 - Let the decreasing series 

 Ci, 2, 83, end with a s = 1. Then 



a; = (- 



F- 1 F- 1 [ (mod p). 

 Li J La 2 J La s _i J 



C. Ladd 53 showed that if a is prime to M = MI -M k and if z* is deter- 

 mined by azi +1 = (mod Mi), the root of ax + b = (mod M) is 



L. Kronecker 54 reduced the solution of az = b (mod w), where a is 

 prime to m = Ilpi', to the case in which m is a power p r of a prime. Then 

 a root can be expressed to the base p in the form 



where each , is an integer chosen from 0, 1, , p 1. First find the root 

 of a = b (mod p). Then seek 1 from a(^ + ip) = b (mod p 2 ), whence 

 ai = (b a )/p (mod p), etc. Again, if JV is the denominator of the 

 next to the last convergent in the continued fraction for a/m, then x = d= bN 

 (mod m). 



M. Lerch 55 showed that, if p is a prime, 



1 10 P ^ \av~\ 



- = a 12 2^ v\ (modp), 



a =i L p J 



where [Jj is the greatest integer = ^, and hence solved ax py = 1. If m 



60 Disq. arith., 1801, Arts. 29, 30; Werke, I, 1863, 20-3; Maser's German transl., 13-15. 



61 Theorie der Congruenzen, in Russian, 1849; in German, 1889, 16, pp. 58-63. 



82 Giornale di Mat., 7, 1869, 115-6. 



83 Math. Quest. Educ. Times, 30, 1879, 41-2. 



84 Vorlesungen iiber Zahlentheorie, 1, 1901, 108-120. 

 M Math. Annalen, 60, 1905, 483. 



