254 report — 1859. 



Report. But Gauss has also suggested a second definition (which is for 

 some purposes slightly more convenient), and which has been adopted by 

 Dirichlet, who defines a primary uneven number to be one in which b is un- 

 even, anda=l, mod 4. The object of singling out one of the four associated 

 numbers is merely that it serves to give definiteness to many theorems. 

 For example, the theorem that " every real number may be expressed as the 

 product of powers of real primes in one way, and in one only," may be now 

 transferred in an equally definite form to the complex theory, " Every complex 

 number can be expressed in one way only in the form i m {\ +i) n A".B' 3 . C^ 

 .... where m, n, a, /3, y, &c. are real integral numbers, A, B, C... primary 

 complex primes." 



If a + bi be a complex number, and N=N(a + fo*)=a 2 +6 2 , and if k be 

 the greatest common divisor of a and b, it can be shown that every number 

 is congruous, for the modulus a + bi, to one, and one only, of the numbers 

 x + iy, where 



N 

 #=0, 1, 2, .... ——1 ; f/=0, 1,2, .... h— 1. 

 h 



These numbers therefore (or any set of numbers congruous to them) form a 

 complete system of residues for the modulus a + bi. The number of the 

 numbers x + iy is evidently N, so that the norm of the modulus represents 

 the number of residues in a complete system. In particular, therefore, if the 

 modulus a + bi be a prime of the second kind, having p for its norm, the num- 

 bers 0, 1, 2,. . .p— 1 represent a complete system of residues ; and if the mo- 

 dulus be a prime of the first kind, as — q, the numbers included in the formula 

 x+iy, where x and y may have any values from to q-~ 1 inclusive, will re- 

 present a complete system of residues. 



26. Fermafs Theorem for Complex Numbers. — Dirichlet's proof of this 

 theorem for ordinary integers is equally applicable to complex numbers, and 

 leads us to the following result : — 



" If p be a prime in the complex theory, and k any complex number not 

 divisible by/?, then k N P- l = \, mod/)." 



Again, the demonstration of the theorem of Lagrange (see Art. 11) is equally 

 applicable here (see Gauss, Theor. Res. Biq., Art. 50), and therefore the 

 general theorems mentioned in Art. 12 may be extended, mutatis mutandis, to 

 the complex theory. In particular, the number of primitive roots will be 

 i^[N(jo) — 1], or the number of numbers less than NQ») — 1, and prime to 

 it. It will follow from this that, if the modulus be an imaginary prime/?, 

 every primitive root of Np in the real theory will be a primitive root both of 

 p and its conjugate. Those Tables of Indices, therefore, in the ' Canon 

 Arithmeticus,' which refer to primes of the form 4w + l will continue to 

 hold, if for the real modules we substitute either of the imaginary factors of 

 which they are composed. For primes of the form 4« + 3 (considered as 

 modules in the complex theory), it would be requisite to construct new tables, 

 — a labour which no one as yet appears to have undertaken. 



27. Law of Quadratic Reciprocity for Complex Numbers. — If p and q be 

 any two uneven primes (not necessarily primary, but subject to the condition 



that their imaginary parts are even), and if we denote by \ — \ the unit-resi- 

 due of the power phl^i-i], mod q ; so that i- 

 p is or is not a quadratic residue of q : then a 

 is expressed by the equation VS. = £ . 



9. 



= +1, or = — 1, according as 



aw of reciprocity exists, which 



