ON THE THEORY OF NUMBERS. 523 
D; for if a=a'o’, & representing the greatest square divisor of a, the equa- 
tion 
e te Dr? — al? 
is certainly resoluble in relatively prime integers, by virtue of a celebrated 
theorem of Legendre* ; and the values of £ which satisfy it are prime to D ; 
peg) ; 
whence, if «= pa »Y=pn, o=p *, p denoting a multiplier, which renders 
the values of x, Ys and w integral and relatively prime, the equation 
ax* + 2bay+cy’=w* will be atisied, and the values of w will be prime to D. 
The form (¢, 6, c) is therefore equivalent to a form of the type (w’, A; v); and 
this form 1s produced by the duplication of (w, A, vw) if w be uneven, and of 
(2w, \+, v') if w be even. 
117. Arrangement of the Classes of the principal Genus.—If C be a 
class of the principal genus, the classes C, C*, C’,. . . will all belong to that 
genus. And it will be found, by reasoning similar to that employed in 
Kuler’s second proof of Fermat’s theorem (see Art. 10 of this Report), (1) 
that the classes 1, C, C?,... are all different until we arrive at a class Cr, 
equivalent to the principal class; (2) that p is either equal to, or a divisor of, 
the number » of classes in the principal genus; (3) that if C’=1, 7 is a mul- 
tiple of ». The p classes C, C*, C®,. . . C+—!, 1, are called the period ¢ of the 
class C; C is said to appertain to the exponent »; and the determinant is 
regular or irregular according as classes do or do not exist which appertain 
to the exponent n. With the former case we may compare the theory of the 
residues of powers for a prime modulus; with the latter the same theory for 
a modulus composed of different primes (see Art. 77). 
(i.) When the determinant is regular, we may take any class appertaining 
to the exponent n as a basis, and may represent all the classes of the principal 
genus (to which we at present confine ourselves) as its powers. It will then 
appear (1) that if d be a divisor of », the number of classes appertaining to 
the exponent d is ¥ (d); so that, for example, the number of classes that 
may be taken for a base is y (n): (2) that if ef=n, the equation X*=1 will 
be satisfied by ¢ classes of the principal genus; and if these classes be repre- 
sented by A,, A,,...A,, each of the equations X/=A will be satisfied by 
f different classes of ‘the same genus: (3) that the only classes of the prin- 
cipal genus which satisfy the equation X*=1 are those which satisfy the 
equation X7=1, where d is the greatest common divisor of & and n. 
It will be seen in particular that the equation X’=1 admits of only one, 
or only two solutions, according as n is uneven or even; 2. é. the principal 
genus of a regular determinant cannot contain more than two ambiguous 
classes. 
To obtain a class appertaining to the exponent n, Gauss employs the same 
method which serves to find a primitive root of a prime number (Art. 13; 
Disq. Arith., art. 73, 74), and which reposes on the observation, that if A 
and B be two classes appertaining to the exponents a and #, neither of which 
divides the other, and if M, the least common multiple of a and #, be re- 
solved into two factors p and q, relatively prime and such that p divides a 
% B 
and q divides 3, the class A? x B@ will appertain to the exponent M. 
(ii.) When the determinant is irregular, the classes of the principal genus 
* Théorie des Nombres, ed. 3, vol. i. p.41; Disq. Arith., art, 294. 
+ These periods of non- equivalent classes are not to be confounded with the periods of 
equivalent reduced forms of Art. 93. 
