795] 
NUMBERS. 
613 
if we please a + br), composed with the roots of this quadric equation,—say the com 
plex numbers a + brj, where a and b are any positive or negative integer numbers, 
including zero. In the case \=23, the quadric equation is 77 2 + 77 + 6 = 0. We have 
N (a + by) = (a + brio) (a + brj^ = a? — ab 4- \ (X + 1) b 2 ; and for X = 23, this is N(a+br)) 
= a 2 — ab -f 6b 2 . 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 X = 23, we have rj 0) tj 1 , the roots of the equation 
V 2 + v + 6 = 0 ; and a real number P is composite when it is = (a 4- brj 0 ) (a + br] 1 ), 
= a 2 — ab + 6b 2 , viz. if 4P = (2a — b) 2 + 23b 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 + 23y 2 ), but some such numbers, residues that is of 23, are composite, for 
instance 59, = (5 — 2?7 0 ) (5 — 2^). And we have an indication, so to speak, of the com 
posite nature of all such numbers; take for instance 13, we have (77 — 4)(77 + 5) =— 2.13, 
where 13 does not divide either 77 — 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 77 + 5. It 
appears, moreover, that a power 13 s is in fact composite, viz. we have 
13 3 = (31 - 12t7 0 ) (31 - I277O, (2197 = 961 + 372 + 864); 
and writing 13 = \/31 — 12t7 0 . v 7 31 - 1277, we have 13 as the product of two ideal 
numbers each represented as a cube root; it is to be observed that, 13 being in the 
simplex theory a prime number, these are regarded as prime ideal numbers. We have 
in like manner 
2 = \/l — 77,,. v / l — 771, 3 = \/l— 2rj 0 . \/l — 2t7j, &c.; 
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) (77 + 5) = — 2.13, or as this may 
be written 
(77, — 4) (77, + 5) = - \/l — rj 0 \/l — 77, v 7 31 — 12770 v 7 31 - 1277J, 
we have (77, — 4) 3 and (1 — 77 0 )(31 — 12t7 0 ) each = 14 + 5577!, or say 
771-4= 1 — 770 31 — 12t7o, 
and similarly 
77!+5=-x/l - vl \/31 — 1277,, 
so that we verify that rjj — 4, 771 + 5, do thus in fact each of them contain an ideal 
factor of 13. 
39. We have 2=v / l— 77,,1 — 771, viz. the ideal multiplier \/l— rj 0 renders actual 
one of the ideal factors vH — 771 of 2, and it is found that this same ideal multiplier