168 REPORT— 1860. 



(by the method given in the last article) deduce the complete solution of 

 x s — 1 = 0, mod p m ~ ■», from that of aP—l==0, mod p ; and then the roots of 

 x^p" — 1=0, mod p m , can be written down at once. 



74. Primitive Boots of the Powers of a Prime. — All the elementary pro- 

 perties of the residues of powers, considered with regard to a modulus which 

 is a power of a prime number, may be deduced from the theorem just proved. 

 In particular, the demonstration of the existence and number of primitive 

 roots (art. 12) is applicable here also; so that we have the theorem: — 



"There are pm-* (/j— 1) \p (j } ~^) residues prime to p m , the successive 

 powers of any one of which represent all residues prime to p m ." These 

 residues are of course the primitive roots of p m . 



If y be a primitive root of p, of the p numbers included in the formula 

 y + hp (mod/) 2 ), p— 1 precisely will be primitive roots of p*. For y+hp 

 is a primitive root of p 2 unless (y + hp)p- 1 =1, modp 2 ; and the congruence 

 ttP-i = 1, mod p 2 , has always one, and only one, root congruous to y for the 

 modulus p. But every primitive root of p" is a primitive root of p 3 , and of 

 every higher power of p, as may be shown by an application of the princi- 

 ple proved in a note to the last article, or, again, by observing that every 

 primitive root of p m + 1 is necessarily congruous, for the modulus p m , to some 

 primitive root of p m , and that there are p times as many primitive roots of 

 pm+\ as f pm m (g ee Jacobi's Canon Arithmeticus, Introduction, p. xxxiii ; 

 also a problem proposed by Abel in Crelle's Journal, vol. iii. p. 12, with 

 Jacobi's answer, ibid. p. 211.) 



75. Case when the Modulus is a Power of 2. — The powers of the even 

 prime 2 are excepted from the demonstrations of the two last articles — in 

 fact, if m > 3, 2'" has no primitive roots. Gauss, however, has shown (Disq. 

 Arith. arts. 90, 91) that the successive powers of any number of the form 

 8ra-f 3 represent, for the modulus I" 1 , all numbers of either of the forms 8ra + 3 

 or 8w + l ; similarly all numbers of the forms 8«+5 and 8n + 1 are repre- 

 sented by successive powers of any number of the form 8n+5. If, there- 

 fore, we denote by y any number of either of the two forms 8h + 3 or 8?i + 5, 

 we may represent all uneven numbers less than 2'" by the formula ( — l)*y' 3 , 



in which a is to receive the values and l,and /3 the values 1, 2,3, 2'"- 2 . 



A double system of indices may thus be used to replace the simple system 

 supplied by a primitive root when sucli roots exist. 



Tables of indices for the powers of 2, and of uneven primes inferior to 

 1000, have been appended by Jacobi to his Canon Arithmeticus. 



76. Composite Modules. — No general theory has been given of the repre- 

 sentation of rational and integral functions of an indeterminate quantity as 

 products of modular functions with regard to a composite modulus divisible 

 by more than one prime. And it is possible that no advantage would be 

 gained by considering the theory of congruences with composite modules 

 from this general point of view. A few isolated theorems relating to par- 

 ticular cases have, however, been given by Cauchy (Comptes Rendus, 

 vol. xxv. p. 26, 1847). Of these the following may serve as a specimen : — 



"If the congruence F (,r) = 0, mod M, admit as many roots as it has 

 dimensions, and if, besides, the differences of these roots Le all relatively 

 prime to M, we have the indeterminate congruence 



F (*)=* (r-rj (.v-r 2 ) (.r-r 3 ) . . . (x-r n ), mod M, 



k denoting the coefficient of the highest power of x in F (.r)." 



But if, instead of considering the modular decomposition of the function 

 F(.r), we confine ourselves to the determination of the real solutions of the 



