ON THE THEORY OP NUMBERS. 163 



The assertion that/(V) is a divisor of F(#), for the modulus p is for 

 brevity expressed by the congruential formula 



¥(x)=0, mod O, /(#)], 



which represents an equation of the form 



¥(x)=pcj>(x)+f(x)^(x). 



Similarly the congruence ¥ l (x)^¥ 2 (x), mod \_p, /(#)], is equivalent to 

 the equation 



F 1 (x) = F 2 (x) + p<p(x)+f(x)4,(x). 

 If f(x) be a function of order m, it is evident that any given function is 

 congruous, for the compound modulus [p,f(x)2 to one > an( ^ one on 'y> °f 

 the p m functions contained in the formula a -\-a 1 x-\- . . . -\-a m —\ x m ~ ', in 

 which a , a v ... « m _i may have any values from zero to p — 1 inclusive. 

 These p m functions, therefore, represent a complete system of residues for 

 the modulus Qp,/(#)]. 



A congruence F(X)=0, mod \_p,f(oc)~], is said to be solved when a func- 

 tional value is assigned to X whicli renders the left-hand member divisible 

 byy(.r) for the modulus p; and the number of solutions of the congruence 

 is the number of functional values (incongruous mod [/>,/(#)]) which may 

 be attributed to X. The coefficients of the powers of X in the function F(X) 

 may be integral numbers or functions of x. The linear congruence AX= B, 

 mod C/>>/(^)], in which A and B denote two modular functions, is, in 

 particular, always resoluble when A is prime tof(x), mod p, and admits, in 

 that case, of only one solution. 



We shall now suppose that the function f(x) in the compound modulus 

 [_Ptf(%)~\ is irreducible for the modulus p, — a supposition which involves the 

 consequence that, if a product of two factors be congruous to zero for the 

 modulus [p,f(%)~], one > at least, of those factors is separately congruous to 

 zero for the same modulus. We thus obtain the principle (cf. art. 11) that 

 no congruence can have more solutions, for an irreducible compound modu- 

 lus, than it has dimensions. For, if X=£, mod [?>,/('#)], satisfy the con- 

 gruence F m (X)=0, mod [p,f(%)~\, we find 



F m (X) = F m (X)-F m (0=(X-OF m _,(X), mod O /(*)], 



F m _ 1 (X) denoting a new function of order m — 1, whence it follows that if 

 the principle be true for a congruence of m — 1 dimensions, it is also true for 

 one of m dimensions; i. e. it is true universally. 



70. Extension of Fer mat's Theorem Let0 denote any one of the p m — 1 



residues of the modulus [_p, f(x)~] which are prime to f(x) ; it may be 

 shown, by a proof exactly similar to Dirichlet's proof of Fermat's theorem, 

 that 



B^-issl, mod O, /(*)] (A) 



This result, which is evidently an extension of Fermat's theorem, involves 

 several important consequences. 



It implies, in the first place, the existence of a theory of residues of powers 

 of modular functions, with respect to a compound modulus, precisely similar 

 to the theory of the residues of the powers of integral numbers with regard 

 to a common prime modulus. A single example (taken from M. Dedekind's 

 memoir) will suffice to show the exact correspondence of the two theories. 

 The modular function 6 is or is not a quadratic residue of f(x), for the 

 modulus p, according as it is or is not possible to satisfy the quadratic con- 

 gruence X 2 =0, mod \_p,f(x)~\. In the former case 6 satisfies the congruence 



m 2 



