ON THE THEORY OF NUMBERS. 167 



73. Binomial Congruences having a Poioer of a Prime for their Modulus. 



If M be any number, and i//(M) represent the number of terms in a system 

 of residues prime to M, it will follow (from a principle to which we have 

 already frequently referred : see arts. 10, 26, 53, 70) that every residue of that 

 system satisfies the congruence a?^W==l, mod M, — a proposition which is 

 well known as Euler's generalization of Fermat's theorem*. In particular, 

 when M=p m , we have xP m ~ x (P-i)sl, mod p m . This congruence has, 

 consequently, precisely as many roots as it has dimensions — a property which 

 is also possessed by every congruence of the form # d =l, mod p m , d denoting 

 a divisor of p m ~' 1 (p—1). This has been established by Gauss in the 3rd 

 section of the Disquisitiones Arithmetical, by a particular and somewhat 

 tedious method f. The simpler and more general demonstration which he 

 intended to give in the 8th section^, was perhaps in principle identical with 

 the following; we exclude the casej»=2, to which indeed the theorem itself 

 is inapplicable : — 



Let d=Sp n , $ representing a divisor of p— 1, and n being ^ m — 1 ; and let 

 us form the indeterminate congruence 



x s — 1 =(#— ej (#—«,) (x— as), mod p m ~ n , 



which is always possible, because x s —l =0, mod p, has 8 incongruous roots. 

 It is readily seen that, if A and B represent two numbers prime to p, and if 

 A=B, mod p r , Ap s =Bp s , modp r + s ; and conversely, if Ap*=Bp", mod^'+«, 

 A = B, modp r §. By applying this principle it may be shown that 



x Spn_ 1 _= r xp n_ a ^ ( pc pn_ a j > n^ ( xp n_ a&p n^ modpt n t 



For if we divide a&>"— 1 by xP n —aP n , the remainder is afP n —\. But, 

 because af = 1 , mod p m ~", a x ¥ l = 1 , mod p m ; i. e.xP n — a/ 71 divides^"— 1 

 for the modulus p m . Similarly x s P n —l is divisible (modp m ) by xP n —aP n 

 etc. ; and since all these divisors are relatively prime for the modulus/?, x s P n — 1 

 is divisible (mod p m ) by their product ; i. e., 



x&p n -\ = (j;P n -a 1 P n ) ( x p n -a./ n ) . . . (xP n —asP n ), mod jo». 



We have thus effected the resolution of x s P n — 1 into factors relatively prime, 

 each of which is congruous (mod p) to a power of an irreducible function ; 

 since evidently (xP n — aP n ) =. (x— a)P n , mod p. To investigate the solutions 

 of x s P n — 1=0, modp m , we have therefore only to consider separately the 

 <S congruences included in the formula xP n = aP n , mod p m . But each of 

 these congruences (by virtue of the principle already referred to) admits 

 precisely p n solutions, viz. the p n numbers (incongruous mod p m ) which are 

 congruous to a, mod p m ~ n . The whole number of solutions of x*P n — 1=0, 

 mod p m , is therefore equal to the index 8p n of the congruence. It further 

 appears that the complete solution of the binomial congruence x i P n —l =0, 

 may be obtained by a direct method, when the complete solution of the 

 simpler congruence x s — 1 =0, modp, has been found. For we may first 



* Euler, Comment. Arith. vol. i. p. 284. 



t Disquisitiones Arithmeticae, arts. 84—88. See also Poinsot, Reflexions sur la Theorie 

 des Nombres, cap. iv. art. 6. 



I Disquisitiones Arithmeticae, art. 84. 



§ If A=B, modpr, but not modj»»-+i, we have A=B+ipr, where k is prime to p. 

 Hence AP*=(B+kp r )P s =BP s +iBP s - 1 /+'+K!> ,+r , K denoting a coefficient divisible 

 hyp ; or AP S ~BP S , mod/ +r , but not modjB s+r+ \ because *BP S_1 is prime to,p. Thi« 

 result implies the principle enunciated in the text. 



