ON THE THEORY OF NUMBERS. 169 



congruence F(.i') = 0, mod M, it 13 always sufficient to consider the con- 

 gruences 



F(.r)=0, mod A, F(.r)=0, mod B, F(»=0, mod C, etc., .... (A) 



where A x B x C . . =M, and A, B, C, . . denote powers of different primes. 

 For if .r = rt, mod A, x = b, mod B, X = c, mod C, denote any solutions of 

 the first, second, third ... of those congruences respectively, it is evident that, 

 if X be a number satisfying the congruences X = a, mod A, X = 6, mod B, 

 X = c, mod C (and such a number can always be assigned), we shall have 

 F(X) =0 for each of the modules A, B, C, . . separately, and therefore for 

 the modulus M; and further, if the congruences (A) admit respectively 

 a, (5, y, ... incongruous solutions, the congruence F(.v)=0, mod M, will 

 admit aX/3x y . . . in all ; for we can combine any solution of F(.r)=0, 

 mod A, with any solution of F(#)=0, mod B, and so on*. 



77. Binomial Congruences with Composite Modules. — The investigation of 

 the real solutions of binomial congruences depends (in the manner just stated) 

 on the investigation of the real solutions of similar congruences the modules 

 of which are the powers of primes. With regard to the relations by which 

 these real solutions are connected with one another, little of importance has 

 been added to the few observations on this subject in the Disquisitiones 

 Arithmeticae (art. 92). If the modulus M=p" q b r c . . . , p, q, r, . . . repre- 

 senting different primes, the congruence iC'J'Wss 1, mod M, possesses no 

 primitive roots; for if n be the least common multiple of p a ~ l (p — ])> 



qb-\ (q—l), r*- 1 (r— 1) n will be less than and a divisor of £ (M). 



But evidently, if x be any residue prime to M, the congruence x n — 1 =0 

 will be satisfied separately for the modules^", q b , r c , . ., and therefore for the 

 modulus M ; i. e., no residue exists, the first t/<(M) powers of which are incon- 

 gruous, mod M. If, however, M=9p a this conclusion does not hold, since 

 the least common multiple of \p (2) and \p (p m ) is ^ (2p"») itself; and we 

 find accordingly that every uneven primitive root of p m is a primitive root of 

 Qp m . When, as is sometimes the case, it is convenient to employ indices to 

 designate the residues prime to a given composite modulus, we must employ 

 (as in the case of a power of 2) a system of multiple indices. To take the 

 most general case, let M=2 9 p" q b r c ..; let u be any number of either of the 

 forms 8w + 3 or 8n + 5, and P, Q, It, . . . primitive roots of p a , q b , r c ,... re- 

 spectively. Then, if n be any given number prime to M, it will always be 

 possible to find a set of integral numbers e n , w n , a n , p„, y n . . . satisfying the 

 conditions 



( - 1 ) e » u">n = n , mod 2* ; < e n < 2, < w„ < 2 e - 2 , 

 P*"~n, mod/>°; 0<a n <p"-i(p— 1), 

 Q"» == n, mod ?* ; 0<fS n <q -1 (q—l), 

 R Y « = n, mod r c ; < X» < r e -i (r— ] ) ; 



• • " • • • • 



and these numbers form a system of indices by which the residue of n for 

 each of the modules 2 e , p a , q b , r c , ... (and consequently for the modulus M) 



* " Infra \ue. in the 8th section] congnientias quascumque secundum modulum e pluribus 

 prmiis compositum, ad congnientias quarum modulus est primus aut primi potestas reducere 

 fusius docebimus " (Disq. Arith. art. 92). It is difficult to sec why Gauss should have em' 

 ployed the word " fusius " if his investigation extended no further than the elementary 

 observations referred to in the text. Nevertheless it is remarkable that Gauss in the 3rd 

 section of the Disq. Arith. sometimes speaks of demonstrations as obscure, which are of 

 extreme simplicity when compared with one in the 4th and several in the 5th scct-'on (%pp in 

 particular arts. 53, 55, 56). v " 



