AND GENERA OE TERNARY QUADRATIC FORMS. 
287 
if therefore the former congruence admits 
primitive solutions, the latter does so too. 
(ii) If the assertion is true for i, 0, it is also true for i, i', where i' <i. 
For if x, y, z is a given primitive solution of 
xz—y 2 =mp\ modp 1 ’, (44) 
Y p l -\-y, Z p'-\-z is a primitive solution of (43), whenever X, Y, Z satisfy the 
congruence 
Xz— 2Yy4-Z^d-^y-^-=m, modj/. 
This congruence admits of pi 21 ' solutions ; for the given numbers x and z are not simul- 
taneously divisible by p. Thus from each primitive solution of (44) we obtain p 2V primi- 
tive solutions of (43). These solutions are all different, and exhaust all the solutions of 
(43) ; if therefore (44) admits of^ 2 ‘ ^1 — solutions, (43) admits of^? 2f+2i '^l — solu- 
tions. 
(iii) The assertion is true if i= 1, i'= 0. For (lemma 1) there are p 2 solutions of the 
congruence xz — y~= 0, modp, and of these one is not primitive. 
The proposition is, therefore, true universally. We shall have to employ the follow- 
ing corollaries from it. 
(1) The function xz — y 2 is divisible by^? 1 , but not by_p i+1 , for p 2 \p — 1) 2 (^ + 1) systems 
of values of x,y , z, mod^? i+1 ; the values of x, y, z not being simultaneously divisible byy?. 
(2) If^> is an uneven prime, the quotients obtained by dividing these J9 2i (^ — 1) 2 (^ + 1) 
values of xz— y 2 by p\ are half quadratic residues, and half non-quadratic residues oip. 
(3) It p=2, the function xz — y 2 is divisible by 2 ! ’, but not by 2 !+1 , for 3 x 2 2i+e systems 
of values of x , y, z , mod 2 i+3 , the values of x, y, z not being simultaneously even. And 
if these 3x2 2i+6 values of xz — y 2 be divided by 2‘, one-fourth part of the quotients is 
contained in each of the linear forms 8&+1, 3, 5, 7. 
Art. 19. Let V = 812x Ilrx ILy x IIS, and let us successively attribute to x,y, z in the 
function xz—y 2 the V 3 systems of values, mod V, of which they are susceptible; let 
p(V) represent the number of those systems, in which the greatest common divisor of 
x, y , z is prime to V, and which give to xz — y 2 a value divisible by 12, and such that the 
quotient is a number of the series M ; if the forms (f) are properly primitive, x 
and z are not to be simultaneously even ; if those forms are improperly primitive, x and 
z are to be simultaneously even. We shall now show that <p(V) is determined by the 
equation 
<p( v)=f x>! x v> x g x n (i -i) x n (i -i) x n (i -A) | 
X n*(i 4)xn(iA,)x n* [ 1 + (=£) I] ’ j 
2 R 
MDCCCLXVII. 
