795] 
NUMBERS. 
603 
The denominator symbol may be negative, say it is — p, we then have as a 
definition 
theorems 
(3\. 
(Q). 
\—p) 
\p) 
-observe that 
zQ 
p 
is not = 
Q 
—p 
-and we have further the 
-1 
P 
viz. — 1 is a residue or a non-residue of p according as p = 1 or = 3 (mod. 4), and 2' 
is a residue or a non-residue of p according as p = 1 or 7, or =3 or 5 (mod. 8). If, 
as 
definitions, ^ ^ j = +1 and = +1, these may be written 
(p 2 —1) 
We have also, what is in fact a theorem given at the end of No. 23, 
(QQ\ 
= (9) 
V p ) 
\p) 
\p) 
The further definition is sometimes convenient— 
= 0, when p divides Q. 
The law of reciprocity, as contained in the theorem 
1)1 (P-D (9-1) > 
(P) 
\q) 
\p) 
is a fundamental theorem in the whole theory; it was enunciated by Legendre, but 
first proved by Gauss, who gave no less than six demonstrations of it. 
5. Jacobis generalized symbol. Jacobi defined this as follows: The symbol 
Q 
, where p, p', p",... are positive odd primes equal or unequal, and Q is any 
±pp'p"...J 
positive or negative odd number prime to pp'p"..., denotes +1 or —1 according to the 
definition 
( Q 
\± pp'p" ••• 
) ip) \p'J 'p") 
the symbols on the right-hand side being Legendre’s symbols. But the definition may 
be regarded as extending to the case where Q is not prime to ppp"...: then we have 
Q divisible by some factor p, and by the definition of Legendre’s symbol in this case 
we have = 0; hence in the case in question of Q not being prime to pp'p"..., 
the value of Jacobi’s symbol is = 0. 
We may further extend the definition of the symbol to the case where the 
numerator and the denominator of the symbol are both or one of them even, and 
present the definition in the most general form, as follows: suppose that p, p, p",... 
76—2