Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/680

NUMBERS where m = 0, or 1, and a, b, c, . . a, /?, y , . are as before. To obtain a like definite statement in the present theory we require to distinguish between the four numbers a 4. 5^ - a - bi, - b + ai, b - ai, which differ from each other only by a unit factor- 1, i. Consider a number a + bi where a and b are the one of them odd and the other even (a and b may be either of them = 0, the other is then odd), every prime number a + bi other than 1 i is necessarily of this form, for if a and b were both even the number would be divisible by 2, or say by (1 + i) 2 , and if a and b were both odd it would be divisible by 1 + i ; then of the four associated numbers a + bi a - bi, b + ai, b - ai, there is one and only one, a + bi, such that b is even and a + b 1 is evenly even ; or say one and only one which is = 1 (mod. 2(1+ i)). We distinguish such one of the four num bers from the other three and call it a primary number ; the units 1, i, and the numbers li are none of them primary numbers. We have then the theorem, a number N is in one way only = i m (l + i) n A a BP . . , where m 0, 1, 2, or 3, n is =0 or a positive integer, A, B, . . . are primary primes, a, (3, . . . positive integers. Here i is a unit of the theory, 1 + i is a special prime having reference to the number 2, but which might, by an extension of the definition, be called a primary prime, and so reckoned as one of the numbers A, B, . . . ; the theorem stated broadly still is that the number N is, and that in one way only, a product of prime factors, but the foregoing complete state ment shows the precise sense in which this theorem must be understood. A like explanation is required in other complex theories ; we have to select out of each set of primes differing only by unit factors some one number as a primary prime, and the general theorem then is that every number N is, and that in one way only, = P . A^SPCT* , . . , where P is a product of unities, and A, B, C, . . are primary primes. 34. We have in the simplex theory (ante, No. 10) the theorem that, p being an odd prime, there exists a system of p - 1 residues, that is, that any number not divisible by p is, to the modulus p, congruent to one, and only one, of the/; - 1 numbers 1, 2, 3, . . .p - 1. The analogous theorem in the complex theory is that for any prime number p other than li there exists a system of N(p}- residues, that is, that every number not divisible by p is, to the modulus _/>, congruent to one of these N(p}- numbers. But p may be a real prime such as 3, or a complex prime such as 3 + 2i ; and the system of residues presents itself naturally under very different forms in the two cases respectively. Thus in the case p = 3, -^(3) = 9, the residues may be taken to be 1,2, i, 1 + i , 2 + i , being in number 7V(3)-1 = 8. And for p = 3 + 2i, 7V(3 + 2i) = 13, they may be taken to be the system of residues of 13 in the simplex theory, viz., the real numbers 1, 2, 3, ..... 12. We have in fact 5 + i = (2 + 3i)(l - i), that is, 5 + i = (mod. 2 + 3i), and consequently a + bi = a- 56, a real number which, when a + bi is not divisible by 3 + 2i, may have any one of the foregoing values 1, 2, 3, ... 12. Taking then any number x not divisible by p, the N"(p) residues each multiplied by x are, to the modulus p, congruent to the series of residues in a different order ; and we thus have, say this is Fermat s theorem for the complex theory x N (^~ l -1=0 (mod. p with all its con sequences, in particular the theory of prime roots. In the case of a complex modulus such as 3 + 2i, the theory is hardly to be distinguished from its analogue in the ordinary theorem ; a prime root is = 2, and the series of powers is 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, for the modulus 3 + 2i as for the modulus 13. But for a real prime such as 3 the prime root is a complex number ; taking it to be = 2 + i, AVC have (2 + i) 8 - 1 = (mod. 3), and the series of powers in fact is 2 + i, i, 2 + 2i, 2,1 + 2i, 2i, l+i,l, viz., we thus have the system of residues (mod. 3). We have in like manner a theory of quadratic residues ; a Legendrian symbol J 1 (which, if p, q, are uneven primes not necessarily primary but subject to the condition that their imaginary parts are even, denotes +1 or - 1 according as pMFi-V is = 1 or = - 1 (mod. q), so that I . I = + 1 or 1 according as p is or is not a residue of q), a law of reciprocity expressed by the very simple form of equation [ J = f - J , and generally a system of properties such as that which exists in the simplex theory. The theory of quadratic forms (a, b, c) has been studied in this complex theory ; the results correspond to those of the simplex theory. 35. The complex theory with the imaginary cube root of unity has also been studied ; the imaginary element is here y, = !( - 1 + f - 3), a root of the equation y 2 + y + 1 = ; the form of the complex number is thus a + by, where a and b are any positive or negative integers, including zero. The conjugate number is a + by 2 , = a-b-by, and the product (a + by)(a + by"), = a 2 -ab + b 2 , is the norm of each of the factors a + by, a + by 2 . The whole theory corresponds very closely to, but is somewhat more simple than, that of the complex numbers a + bi. 36. The last -mentioned theory is a particular case of the complex theory for the imaginary Ath roots of unity, A being an odd prime. Here a is determined by the equation a x - 1 - = 0, that is, a x ~ * + a x ~ 2 + ... + a + 1 = 0, and the f orm of the complex number is/(a), = a + ba + ca 2 . . . + ia* ~ 2 > where a, b, c, . . . k, are any positive or negative integers, including zero. We have A - 1 conjugate forms, viz., /(a), /(a 2 ), .... /(a x ~ *), and the product of these is the norm of each of the factors Ay (a), = Nf(o? ...= A/ (a x " *). Taking g any prime root of A, g^ ^-l = (mod. A), the roots a, a 2 , . . . a x ~ *, may be arranged in the order a, a$, o$ 2 , . . . a ( J ~ 2 ; and we have thence a grouping of the roots in periods, viz., if A - 1 be in any manner whatever expressed as a product of two factors, A - 1 = ef, we may with the A- 1 roots form e periods TJ O , ^, . . . >7<>_i, each of / roots. For instance A = 1 3 ; a prime root is g = 2, and A-l=?/=3.4; then the three periods each of four roots are w = a +a 8 + a 12 + a 5 , So also if ef= 2 . 6 then the 2 periods each of 6 roots are ?7 = a +a 4 + a 3 + a 1 - + a 9 + a 10 , 7] L = a 2 + a 8 + a 6 + a 11 + a 3 + a 7 ; and so in other cases. In particular, if /= 1 and conse quently e = -l, the e periods each of / roots are in fact X-2 the single roots a, off, . . . aO ". We may in place of the original form of the complex number consider the new form /(^) = ^ + bi^ . . . + ^ e _ 1? which when / = 1 is equivalent to the original form, but in any other case denotes a special form of complex number ; instead of A - 1 we have only e conjugate numbers, and the product of these e numbers may be regarded as the norm of f(i]). 37. The theory for the roots a includes as part of itself the theory for the periods corresponding to every decom position whatever A - 1 = </ of A-l into two factors, but