ADDRESS. 33 
throughout, a theory of ordinary real numbers. It establishes the notion 
of a congruence ; gives a proof of the theorem of reciprocity in regard to 
quadratic residues; and contains a very complete theory of binary quadratic 
forms (a, b, c)(#, y)?, of negative and positive determinant, including the 
theory, there first given, of the composition of such forms. It gives also 
the commencement of a like theory of ternary quadratic forms. It con- 
tains also the theory already referred to, but which has since influenced 
in so remarkable a manner the whole theory of numbers—the theory of 
the solution of the binomial equation «” — 1=0: it is, in fact, the roots 
or periods of roots derived from these equations which form the incom- 
measurables, or unities, of the complex theories which have been chiefly 
worked at; thus, the 7 of ordinary analysis presents itself as a root of 
the equation 2t—1=0. It was Gauss himself who, for the develop- 
ment of a real theory—that of biquadratic residues—found it necessary 
to use complex numbers of the before-mentioned form, a + bi (a and D 
positive or negative real integers, including zero), and the theory of these 
numbers was studied and cultivated by Lejeune-Dirichlet. We have thus 
a new theory of these complex numbers, side by side with the former 
theory of real numbers : everything in the real theory reproducing itself, 
prime numbers, congruences, theories of residues, reciprocity, quadratic 
forms, &c., but with greater variety and complexity, and increased diffi- 
culty of demonstration. But instead of the equation at — 1 = 0, we may 
take the equation «3 -1=0: we have here the complex numbers 
a+ bp composed with an imaginary cube root of unity, the theory 
specially considered by Eisenstein: again a new theory, corresponding 
to but different from that of the numbers a + bi. The general case of 
any prime value of the exponent n, and with periods of roots, which here 
present themselves instead of single roots, was first considered by Kum- 
" mer: viz. ifn —1=ef,and n,,n. ... n, are thee periods, each of them 
a sum of f roots, of.the equation «” — 1 = 0, then the complex numbers 
considered are the numbers of the form «a %}, + uy Ng i bay 
(a, @... a, positive or negative ordinary integers, including zero) : 
f may be = 1, and the theory for the periods thus includes that for the 
single roots. 
We have thus a new and very general theory, including within itself 
that of the complex numbers a+bianda+bp. But anew phenomenon 
presents itself; for these special forms the properties in regard to prime 
numbers corresponded precisely with those for real numbers ; a non-prime 
number was in one way only a product of prime factors ; the power of a 
prime number has only factors which are lower powers of the same prime 
number : for instance, if p be a prime number, then, excluding the obvious 
decomposition p. p?, we cannot have p= a product of two factors A, B. 
Tn the general case this is not so, but the exception first presents itself for 
the number 23; in the theory of the numbers composed with the 23rd roots 
of unity, we have prime numbers p, such that p>=AB. To restore the 
theorem, it is necessary to establish the notion of ideal numbers ; a prime 
1888. D 
