ON THE THEORY OP NUMBERS. 253 



bers, we have evidently N(a)x N(/3) = N(a/3). There are in this theory 

 four units, 1, i, — 1, — i, which have each of them a positive unit for their 

 norm. The four numbers a-\-bi, ia—b, —a—ib, — ia + b (which are obtained 

 by multiplying any one of them by the four units in succession, and which 

 consequently stand to one another in a relation similar to that of + a and — a 

 in the real theory) are said to be associated numbers. These four associated 

 numbers with the numbers respectively conjugate to them form a group of 

 eight numbers (in general different), all of which have the same norm. These 

 definitions are applicable whatever be the nature of the real quantities a and b. 

 If a and b are both rational, the complex number is said to be rational ; if 

 they are both integers, a + bi is a complex integral number. One complex 

 integer a is said to be divisible by another ft, when a third y can be found 

 such that a=/3y. Adopting these definitions, Ave can show that Euclid's pro- 

 cess for investigating the greatest common divisor of two numbers is equally 

 applicable to complex numbers; for it may be proved that, when we divide 

 one complex number by another, we may always so choose the quotient as 

 to render the norm of the remainder not greater than one-half of the norm 

 of the divisor*. If, therefore, we apply Euclid's process for finding the 

 greatest common divisor to two complex numbers, we shall obtain remainders 

 with norms continually less and less, thus at last arriving at a remainder 

 equal to zero ; and the last divisor will be, as in common arithmetic, the 

 greatest common divisor of the two complex numbers. Similarly the funda- 

 mental propositions deducible in the case of ordinary integers from Euclid'3 

 theory are equally deducible from the corresponding process in the case of 

 complex integral numbers. Thus, " if a complex number be prime to each of 

 two complex numbers, it is prime to their product." " If a complex number 

 divide the product of two factors, and be prime to one of them, it must 

 divide the other." "The equation ax — by=\, where a and b are complex 

 numbers prime to one another, is always resoluble with complex numbers 

 x and y, and admits an infinite number of solutions," &c. 



A prime complex number is one which admits no divisors besides itself, its 

 associates, and the four units. 



There are three distinct classes of primes in the complex theory : — 



1. Real prime numbers of the form 4w + 3 (with their associates). 



2. Those complex numbers whose norms are real primes of the form 4«-f 1. 



3. The number 1 +i and its associates the norm of which is 2. 



Instead of dividing numbers into even and uneven, we must here divide 

 them into three classes, uneven, semi-even, and even, according as they are 

 (1) not divisible by (l+«); (2) divisible by 1+i, but not by (l+«') 2 5 ( 3 ) 

 divisible by (1 + i) 2 =2i, or, which is the same thing, by 2. 



Of four associated uneven numbers, there is always one, and only one, such 

 that b is even and a + b — 1 evenly even. This is distinguished from the others 

 as primary. Thus — 7 and — 5 + 2i are primary numbers. A primary 

 number is congruous to + 1 for the modulus 2(1 + 2); whence it appears 

 that the product of any number of primary numbers is itself a primary 

 number. The conjugate of a primary is also primary. In speaking of un- 

 even numbers, unless the contrary is expressed, we shall suppose them to be 

 primary. This definition of a primary number is that adopted by Gauss (I.e. 

 Art. 36), and after him by Eisenstein, and we shall adhere to it in this 



* Since 2+^ = a 4±^ + b -p°°! i ; if /> be the integral number nearest to ™+M, and 

 c-\-di c+a 2 c^+rf 3 c+d* 



q that nearest to s ~ , p+qi is the quotient required. 



