ON THE THEORY OK NUMBERS. 139 



(<t.) If /(a) denote any (actual) complex number prime to X (i. e. not 

 divisible by 1— a), a complex unit e (a) can always be assigned, such that 

 the product F(a)=e (a)f(a) shall satisfy the two congruences 

 F («) F (*-') = [F (l)] 2 , mod X, 

 F(*) = F(1), mod(l-a) 2 . 

 A complex number satisfying these two congruential conditions is called a 

 primary complex number ; the product of two primary numbers is there- 

 fore itself primary. This definition, in the particular case X=3, includes 

 the primary numbers of art. 37, taken either positively or negatively. 



53. Fermat's Theorem for Complex Primes. — Let <p (a) be an actual or 

 ideal complex prime, and let N = N . cp (a) represent its norm. A system of 

 N actual numbers can always be assigned such that every complex number 

 shall be congruous to one and only to one of them for the modulus cj> (a). 

 These N numbers may be said therefore to form a complete system of 

 residues for the modulus $(a) ; and by omitting the term divisible by (j> («), 

 we obtain a system of N — I residues prime to <j> (a). 



Let q be a prime appertaining to the exponent/, so that N=g , / ) and let 

 <p (a) or (p x (»jo) be the prime factor of q which appertains to the substitution 

 jj =m ; the formula 



a + a l a-\-a 3 a 2 + . .. +a/-i a-f- 1 , (A) 



will represent a complete system of residues for the modulus (j> 1 (»/o)> if we 

 assign to the coefficients a , a v a 2 .... the values 0, 1,2, ...q — 1, in succession. 



For if/(*)=^o('/o) + a '/'i (»/o) + .. + ... *f- 1 i///_i (i/ ) be any complex num. 

 ber,/(a) is congruous for the modulus <p t (j/ ) to \\i (u ) + cc\pi (m ) +. .-\-af~ 1 

 \|//_i(m )j because Mo — r} = 0,mod(b 1 (r] ); that is, /'(a) is congruous to one 

 of the complex numbers included in (A); nor can any two numbers a + 

 a l a + a 2 a 2 -)- . . + a/-i a.f~ x and b + b 1 a.+b % a 2 + . . . + b/-\ a.f~ l included 

 in that formula be congruous to one another ; for the congruence (a — #o) + 

 a(a 1 —b 1 ) + a 2 (a 2 —b 2 ) + ...+ a./- 1 (a/-i— 4/_i) = 0, mod ^ (?j ), involves, 

 by M. Rummer's theory (see art. 45 ), the coexistence of the/ congruences 

 a — &o = 0, mod q; a l —b l ^Q, mod q; . . .a/_i— 6/_i =0, mod q; i. e. the 

 identity of the complex n umbers a + a.a l + a 2 a 2 + . . .a/- 1 a/_ ] ,and6 + a& 1 + 

 a 2 6 2 +. . + a/ _1 bf-\. It is worth while to notice that, if q be a prime ap- 

 pertaining to the exponent 1, for the modulus X, i. e. if q be of the linear form 

 wiX + 1, the real numbers 0, 1, 2, 3...q— 1 will represent the terms of a 

 complete system of residues for the modulus <p (a) ; but if <p (a) be a factor 

 of a prime appertaining to any higher exponent than unity, a complete system 

 will contain complex as well as real integral residues. 



By applying the principle (see art. 10) that a system of residues prime 

 to the modulus, multiplied by a residue prime to the modulus, produces 

 a system of residues prime to the modulus, we obtain the theorem, which 

 here replaces Fermat's Theorem, that if i^(a) be any actual number prime 

 to <f) («), [;// (a)] 1 *- 1 = 1, mod</> (a). If we combine with this theorem the 

 principle of Lagrange (cited in art. 11) which is valid for complex no less 

 than for real prime modules, we may extend, mutatis mutandis, to the general 

 complex theory the elementary propositions relating to the Residues of 

 Powers, Primitive Roots, and Indices, which, as we have seen, exist in the 

 case of complex primes formed with cubic or biquadratic roots of unity. In 

 fact, these propositions are of a character of even greater generality, and may 

 be extended, not only to complex numbers formed with roots of unity whose 

 index is a composite number, but also to all complex numbers formed with 

 the roots of equations having integral coefficients, as soon as the prime fac- 

 tors of those complex numbers are properly defined. 



