602 
NUMBERS. 
[795 
23. Quadratic residues for an odd prime modulus. In particular, if h = 2, then 
e — 2, f= i (p — 1), and the square of every number not divisible by p is = one of the 
z(p— 1) numbers g~, g 4 , that is, there are only ^(p-1) numbers out of the 
series 1, 2, 3, ...,p — 1 which are residues of a square number, or say quadratic residues, 
and the remaining \ (p — 1) numbers are said to be quadratic non-residues of the 
modulus p,—we may say simply, residues and non-residues. But this result can be 
obtained more easily without the aid of the theory of prime roots. Every number not 
divisible by p is, to the modulus p, = one of the series of numbers +1, ±2, +3, ..., ±\{p—1); 
hence every square number is = one of the series of numbers l 2 , 2 2 , 3 2 , ..., \{p — l) 2 ; 
and thus the p — 1 numbers 1, 2, 3, p — 1, are one-half of them residues and the 
other half non-residues of p. Thus, in the case p = 11, every number not divisible by 
11 is, to this modulus, = one of the series +1, ±2, +3, + 4, + 5 ; whence the square 
of any such number is = one of the series 1, 4, 9, 16, 25, or say the series 1, 4, 9, 5, 3; 
that is, we have 
residues 1, . 3, 4, 5, . . . 9, . 
non-residues . 2, . . . 6, 7, 8, . 10 
Calling as usual the residues a and the non-residues b, we have in this case 
-&(Xb -2a) =^(33 -22), =1, 
a positive integer; this is a property true for any prime number of the form 4n + 3, 
but for a prime number of the form 4n + 1 we have Xb — Xa — 0 ; the demonstration 
belongs to a higher part of the theory. 
It is easily shown that the product of two residues or of two non-residues is a 
residue; but the product of a residue and a non-residue is a non-residue. 
24. The law of reciprocity—Legendre’s symbol. The question presents itself, given 
that P is a residue or a non-residue of Q, can we thence infer whether Q is a 
residue or a non-residue of P ? In particular, if P, Q, are the odd primes p, q, for 
instance, given that 13 = 12(17), can we thence infer that 17 = R (13), or that 
17 = JS T R (13) ? The answer is contained in the following theorem: If p, q, are odd 
primes each or one of them of the form 4n -f 1, then p, q, are each of them a residue 
or each of them a non-residue of the other ; but, if p, q, are each of them of the 
form 4n + 3, then, according as p is a residue or a non-residue of q, we have q a non 
residue or a residue of p. 
The theorem is conveniently expressed by means of Legendre’s symbol, viz. p being 
a positive odd prime, and Q any positive or negative number not divisible by p, then 
(p} ^ eno ^ es 1 or ~ L according as Q is or is not a residue of p; if, as before, q 
is (as p) a positive odd prime, then the foregoing theorem is