ON THE THEORY OF NUMBERS, 779 
that the absolute value of w is less than p, so that x is not completely deter- 
mined, but is either the least positive or the least negative residue of the 
binomial coefficient ; though in this case if A is a prime of the form 4n+3, 
the ambiguity may be removed by the congruence 3 #=1,mod A. But 
when the exponent of p is > 2, x is never completely determined by the 
congruence for the modulus p. 
It is very remarkable that the exponent of p in the formula (A) is pre- 
cisely the number of improperly primitive classes of determinant —A, and 
in the formula (B) is precisely the number of properly primitive classes of 
determinant —A*. 
Before Dirichlet’s discovery of the formule expressing the number of 
classes of quadratic forms of a given determinant, Jacobi, having succeeded 
in determining the exponent of p in the formula (A), for the case in which 
A is a prime number, was led with singular sagacity to conjecture that 
pb— Ta 
A 
must represent the number of improperly primitive classes of deter- 
* See Art. 104 of this Report. When A is of the form 4x+3, the two expressions given 
by Dirichlet for the number of properly primitive classes of determinant —A are 
( 2-(5)) =) , and A—B, where A and B represent the numbers of residues infe- 
tisfying the conditions (“~)=+1 and ()=—1 respectively, Hence 
rior to 3A, and satisfying the conditions (4) +1 an (7) respectively en 
#aE is the number of improperly primitive classes; because that number is equal to 
or is one-third of the number of properly primitive classes, according as A==7, or=3, 
mod 8 (see Art. 103 or 113). 
There is no difficulty in showing that Dirichlet’s two expressions are identical. If 
() =1, the congruence 24'=4, mod A, is always resolubie; and if } receive in succession 
all positive values less than A which satisfy the condition (¢)=-1 b! will obtain the 
=o! b 26'—b 
i. eet ee A But if b'<iA, 
same values in a different order. Hence = a =2 
2b'—b=0; if b'>+4A, 20'—0=A, 7. ¢. ne =A, for there are A values of 4 greater than 3A. 
>b— Sa 
Similarly =o =B, so that =A-B. In precisely the same manner it may be 
shown, if ‘@re by considering the congruences 2b'+-5=0, mod A, 2a’+a=0, 
2b'+6 2a' +a 
A 
x =2B+A; whence 
33 
mod A, that = => =2A+B and — => 
3 2-24 _a_p, 
A 
Also the expression given in Art. 104 coincides with A—B. For that expression may be 
written in the form 
-OrC2)-2C2)-@-+@-- 
a' and 0! representing numbers less than 34. 
m[F(@—“")}2 
UE Zany , the exponent of p in the 
Tf we consider, as Cauchy has done, the product 
nLF67)) 
formula (A) will be A—B. That product is evidently equal to TF(@—™), or to nO) , 
