Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/681

Rh NUMBERS 623 each of these may be treated apart from the others as a theory complete in itself. In particular, a simple case is that of the half -periods e = 2, f= |(A - 1) ; and, inasmuch as the characteristic phenomenon of ideal numbers presents itself in this theory of the half-periods (first for the value A = 23), it will be sufficient, by way of illustration of the general theory, to consider only this more special and far easier theory ; we may even assume A = 23. l For the case in question, A - 1 = ef= 2. |(A - 1), we have the two periods r/, 77^ each of ^(A-l) roots; from the expressions for y, y v in terms of the roots we obtain at once ^ + ^ x = 1, and with a little more difficulty &amp;gt;/ 7/ 1 =-|(A-l) or |(A+1) according as A is = 1 or 3 (mod. 4), that is, in the two cases respectively ?7, r) v are the roots of the equation 772 + 77 - ^(A- 1) = 0, and f) 2 + f + ^( A + 1 ) = 0. And this equation once obtained there is no longer any occasion to consider the original equation of the order A - 1, but the theory is that of the complex numbers ar) + br) v or if we please a + 677, composed with the roots of this quadric equation, say the complex numbers a + by, where and b are any positive or negative integer numbers, including zero. In the case A = 23 the quadric equation is 772 + 77 + 6 = 0. We have N(a + brj) = (a + br) Q )(a + b^) = a 2 - ab + ^(A + 1)6 2 ; and for A = 23 this is N(a + brj) = a 2 - ab + 66-. It may be remarked that there is a connexion with the theory of the quadratic forms of the determinant - 23, viz., there are here the three improperly primitive forms (2, 1, 12), (4, 1, 6), (4, - 1, 6), 23 being the smallest prime number for which there exists more than one improperly primitive form. 38. Considering then the case A = 23, we have %, 77^ the roots of the equation ?? 2 + 77 + 6 = ; and a real number P is composite when it is = (a + br) )(a + btjj), =ct?-ab + 66 2 , viz., if 4P = (2a-&) 2 + 236 2. Hence no number, and in particular no positive real prime P, can be composite unless it is a (quadratic) residue of 23 ; the residues of 23 are 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18; and we have thus, for instance, 5, 7, 11, as numbers which are not composite, while 2, 3, 13, are numbers which are not by the condition precluded from being composite : they are not, according to the foregoing signification of the word, composite (for 8, 12, 52, are none of them of the form x 2 + 23f), but some such numbers, residues that is of 23, are composite, for instance 59, = (5 - 2r? )(5 - 2^). And we have an indi cation, so to speak, of the composite nature of all such num bers ; take for instance 1 3, we have (77 - 4)(*7 + 5)= -2.13, where 1 3 does not divide either 17 - 4 or 77 + 5, and we are led to conceive it as the product of two ideal factors, one of them dividing 77 - 4, the other dividing t] + 5. It appears, moreover, that a power 13 3 is in fact composite, viz., we have 13 3 = (31 - 12r, )(31 - 127,!), (2197 = 961 + 372 + 864) ; and writing 13 = /31 12r/. 4/31 - 12r/ 1 we have 13 as the product of two ideal numbers each represented as a cube root ; it is to be observed that, 1 3 being in the sim plex theory a prime number, these are regarded as prime ideal numbers. We have in like manner 1 In the theory of the roots a, ideal numbers do not present them selves for the values X = 3, 5, or 7 ; they do for the value X = 23. It is stated in Smith s &quot;Report on the Theory of Numbers,&quot; Brit. Assoc. Report for 1860, p. 136, that &quot; for the intermediate cases X = ll, 13, 17, and 19, it is uncertain whether they do or do not present them selves.&quot; The writer is not aware whether this question has been settled ; but in Reuschle s Tafeln, 1875, no ideal factors present them selves for these values of X ; and it is easy to see that in the theory of the half-periods the ideal factors first present themselves for the value X = 23. It may be remarked that the solution of the question depends on the determination of a system of fundamental units for the values in question X = ll, 13, 17, and 19 ; the theory of the units in the several complex theories is an important and difficult part of the theory, not presenting itself in the theory of the half-periods, which is alone attended to m the text. every positive real prime which is a residue of 23 is thus a product of two factors ideal or actual. And, reverting to the equation (77 - 4)(q + 5) = - 2. 13, or as this may be written we have fo - 4)3 and (1 - % )(31 - 1277 ) each = 14 + 55r,,, or say and similarly so that we verify that ^ - 4, ^ + 5, do thus in fact each of them contain an ideal factor of 13. 39. We have 2=^/1-^ v/1 - ij lt viz., the ideal multi- plier 4/1 - ?7 renders actual one of the ideal factors 2, and it is found that this same ideal multi renders actual one of the two ideal factor^ /l 77 ; of plier /l - of any other decomposable number 3, 13, etc., /31 - 127,0 Similarly the conjugate multiplier ^= -5-7,, &c. ^ renders actual the other ideal factor of any number 2, 3, 13, &c. We have thus two classes, or, reckoning also actual numbers, three classes of prime numbers, viz., (1) ideal primes rendered actual by the multiplier /l-77 , (2) ideal primes rendered actual by the multiplier 4/1 - 77^ (3) actual primes. This is a general property in the several complex theories ; there is always a finite number of classes of ideal numbers, distinguished according to the multipliers by which they are rendered actual ; the actual numbers form a &quot;principal&quot; class. 40. General theory of congruences irreducible func tions. In the complex theory relating to the roots of the equation 77 2 + 77 + 6 = there has just been occasion to consider the equation (77 - 4)(7; + 5) = -2.13, or say the congruence (77 - 4)(77 + 5) = (mod. 13) ; in this form the relation tf + 77 + 6 = is presupposed, but if, dropping this equation, 77 be regarded as arbitrary, then there is the congruence ff + rj + 6 = (77 - 4)(77 + 5) (mod. 13). Fora different modulus, for instance 11, there is not any such congruence exhibiting a decomposition of 772 + 77 + 6 into factors. The function 772 + 77 + 6 is irreducible, that is, it is not a product of factors with integer coefficients; in respect of the modulus 13 it becomes reducible, that is, it breaks up into factors having integer coefficients, while for the modulus 11 it continues irreducible. And there is a like general theory in regard to any rational and integral function F(x) with integer coefficients ; su/:h function, assumed to be irreducible, may for a given prime modulus p continue irreducible, that is, it may not admit of any decomposition into factors with integer coefficients ; or it may become reducible, that is, admit of a decomposition F(x) = tf&amp;gt;(x)fa;)(x) . . (mod. p). And, when this is so, it is thus a product, in one way only, of factors &amp;lt;(.*), 4&amp;lt;(x), x(^)) ? which are each of them irreducible in re gard to the same modulus p ; any such factor may be a linear function of x, and as such irreducible ; or it may be an irreducible function of the second or any higher degree. It is hardly necessary to remark that in this theory func tions which are congruent to the modulus p are regarded as identical, and that in the expression of F(x) an irre ducible function &amp;lt;J&amp;gt;(x) may present itself either as a simple factor, or as a multiple factor, with any exponent. The decomposition is analogous to that of a number into its prime factors ; and the whole theory of the rational and integral function F(x) in regard to the modulus p is in many respects analogous to that of a prime number re garded as a modulus. The theory has also been studied where the modulus is a power p v.