Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/673

NUMBERS 615 In the ordinary theory we have, in the first instance, positive integer numbers, the unit or unity 1, and the other numbers 2, 3, 4, 5, &c. We introduce the zero 0, which is a number sui generis, and the negative numbers - 1, -2, - 3, -4, &c., and we have thus the more general notion of integer numbers, 0, 1, 2, 3, tfcc. ; + 1 and - 1 are units or unities. The sum of any two or more numbers is a number ; conversely, any number is a sum of two or more parts ; but even when the parts are positive a number cannot be, in a determinate manner, represented as a sum of parts. The product of two or more numbers is a number ; but (disregarding the unities +1, - 1, which may be introduced as factors at pleasure) it is not con versely true that every number is a product of numbers. A number such as 2, 3, 5, 7, 11, &c., which is not a product of numbers, is said to be a prime number ; and a number which is not prime is said to be composite. A number other than zero is thus either prime or composite ; and we have the theorem that every composite number is, in a deter minate way, a product of prime factors. We have complex theories in which all the foregoing notions (integer, unity, zero, prime, composite) occur; that which first presented itself was the theory with the unit i (i 2 - 1) ; we have here complex numbers, a + bi, where a and b are in the before-mentioned (ordinary) sense positive or negative integers, not excluding zero ; we have the zero 0, = + Oi, and the four units 1, - 1, i, -i. A number other than zero is here either prime or else composite; for instance, 3, 7, 11, are prime numbers, and 5, = (2 + *)(2 - i), 9, = 3 . 3, 13, = (3 + 2i)(3 - 2t), are com posite numbers (generally any positive real prime of the form 4n + 3 is prime, but any positive real prime of the form 4 + 1 is a sum of two squares, and is thus composite). And disregarding unit factors we have, as in the ordinary theory, the theorem that every composite number is, in a determinate way, a product of prime factors. There is, in like manner, a complex theory involving the cube roots of unity if a be an imaginary cube root of unity (a 2 + a + 1 = 0), then the integers of this theory are a + ba, (a and b real positive or negative inte gers, including zero) a complex theory with the fifth roots of unity if a be an imaginary fifth root of unity (a 4 + a 3 + a 2 + a + 1 = 0), then the integers of the theory are a + ba + ca 2 + da? (a, b, c, d, real positive or negative integers, including zero) ; and so on for the roots of the orders 7, 11, 13, 17, 19. In all these theories, or at any rate for the orders 3, 5, 7 (see No. 37, post], we have the foregoing theorem : disregarding unit factors, a number other than zero is either prime or composite, and every composite number is, in a determinate way, a product of prime factors. But coming to the 23d roots of unity the theorem ceases to be true. Observe that it is a particular case of the theorem that, if ^V be a prime number, any integer power of N has for factors only the lower powers of N, for instance, N Z = N. N 2 ; there is no other decomposition A f3 = AB. This is obviously true in the ordinary theory, and it is true in the complex theories preceding those for the 3d, 5th, and 7th roots of unity, and probably in those for the other roots preceding the 23d roots ; but it is not true in the theory for the 23d roots of unity. We have, for instance, 47, a number not decomposable into factors, but 47 3 , = AB, is a product of .two numbers each of the form a + ba + . . + ka zi (a a 23d root). The theorem recovers its validity by the introduc tion into the theory of Rummer s notion of an ideal number. The complex theories above referred to would be more accurately described as theories for the complex numbers involving the periods of the roots of unity : the units are the roots either of the equation x p ~ l + x p ~ * . . . + x+l =0 -1 (p a prime number) or of any equation x e +..1=0 belonging to a factor of the function of the order p - 1 (in particular, this may be the quadric equation for the periods each of (p - 1) roots); they are the theories which were first and have been most completely considered, and which led to the notion of an ideal number. But a yet higher generalization which has been made is to consider the complex theory, the units whereof are the roots of any given irreducible equation which has integer numbers for its coefficients. There is another complex theory the relation of which to the foregoing is not very obvious, viz., Galois s theory of the numbers composed with the imaginary roots of an irreducible congruence, F(x) = (modulus a prime number 2)) ; the nature of this will be indicated in the sequel. In any theory, ordinary or complex, we have a first part, which has been termed (but the name seems hardly wide enough) the theory of congruences ; a second part, the theory of homogeneous forms (this includes in particular the theory of the binary quadratic forms (a, b, c)(x, y) 2 ) ; and a third part, comprising those miscellaneous investiga tions which do not come properly under either of the fore going heads. Ordinary Theory, First Part. 1. We are concerned with the integer numbers 0, 1, 2, 3, &c., or in the first place with the positive integer numbers 1, 2, 3, 4, 5, 6, <fec. Some of these, 1, 2, 3, 5, 7, &c., are prime, others, 4, = 2 2 , 6, = 2 . 3, &c., are composite ; and we have the fundamental theorem that a composite number is expressible, and that in one way only, as a pro duct of prime factors, N=a a bPtf. . . (a, b, c, .. primes other than 1 ; a, (3, y, . . positive integers). Gauss makes the proof to depend on the following steps : (1) the product of two numbers each smaller than a given prime number is not divisible by this number ; (2) if neither of two numbers is divisible by a given prime number the product is not so divisible ; (3) the like as regards three or more numbers ; (4) a composite number cannot be resolved into factors in more than one way. 2. Proofs will in general be only indicated or be altogether omitted, but, as a specimen of the reasoning in regard to whole numbers, the proofs of these fundamental propositions are given at length. (1) Let/> be the prime number, a a number less than p, and if possible let there be a number b less than p, and such that ah is divisible by p ; it is further assumed that b is the only number, or, if there is more than one, then that b is the least number having the property in question ; b is greater than 1, for a being less than p is not divisible by p. Now p as a prime number is not divisible by b, but must lie between two consecutive multiples mb and (m+l)b of b. Hence, ab being divisible by p, mab is also divisible by p ; moreover, a}) is divisible byp, and hence the difference of these numbers, a(p- mb), must also be divisible by p, or, writing p - mb = b , we have ab divisible by p, where b is less than b ; so that b is not the least number for which ab is divisible by/>. (2) If a and b are neither of them divisible by;?, then a divided )jp leaves a remainder a which is less than p, say we have a = mp + a ; and similarly b divided by p leaves a remainder ft which is less than p, say we have b = np + /3 ; then ab = (mp + a)(np + /3), = (mnp + na + m?) p + a/3, and a/3 is not divisible by p, therefore ab is not divisible by p. (3) The like proof applies to the product of three or more factors a, b, c, . . . (4) Suppose that the number jyr = a a bPcy . . . (a, b, c, . . . prime numbers _ other than 1) is decomposable in some other way into prime factors ; we can have no prime factor p, other than a, b, c, . . . , for no such number can divide aPlrc * . . . ; and we must have each of the numbers a, b, c, . . . , for if any one