Notes on Quadratic Residues
Definition. Let p be an odd prime, n be an integer with p | n . If congruence
x 2  n (mod p) has a solution we say that n is a quadratic residue mod p.
Otherwise, we say that n is a quadratic nonresidue mod p.
Theorem. Let p be an odd prime. Then every reduced residue system mod p contains
exactly ( p  1) / 2 quadratic residues and exactly ( p  1) / 2 quadratic
nonresidues mod p. The quadratic residues belong to the residue classes
containing the numbers
12 , 2 2 ,  , (
p 1 2
) .
2
Legendre’s symbol and its properties
n
Definition. Let p be an odd prime. We define Legendre’s symbol   as follows:
 p
n
If n is quadratic residue mod p,    1 .
 p
n
If n is quadratic nonresidue mod p,    1 .
 p
n
If n  0 (mod p) ,    0 .
 p
n
Remark: Some authors write (n | p ) instead of   .
 p

1
   1 for any odd prime p.
 p

a b
     if a  b (mod p ).
 p  p
Page 1 of 4

(Euler Criterion) Let p be odd prime. Then for all n we have
n
   n
 p
p1
2
(mod p ) .
p 1
 1
  1, p  1 (mod 4)
In particular, we have    (1) 2  
 p
  1, p  3 (mod 4)


n
Legendre’s symbol   is a completely multiplicative function of n. That
 p
means for any integers a, b we have
 ab   a   b 
        .
 p   p  p
For every odd prime p, we have
2
   (1)
 p

p 2 1
8
  1,

  1,
p  1 (mod 8)
p  3 (mod 8)
(Quadratic reciprocity law) If p and q are distinct odd primes, then
 p  q 
    
 q   p
except when p  q  3 (mod 4) , in which case
 p
q
    
q
 p
( p 1)( q 1)
 p  q 
Equivalently,       (1) 4
for all distinct odd primes p and q.
 q   p
Example: Is 83 a quadratic residue mod 103?
2
 83 
 103 
 20 
 2  5
 5
 83 
3
Solution: 
  
                   1 .
 103 
 83 
 83 
 83   83 
 83 
 5
5
So 83 is a quadratic residue mod 103.
Page 2 of 4
Jacobi symbol and its properties
r
Definition. If P is a positive odd integer with prime factorization P   piai . The
i 1
n
Jacobi symbol   is defined for all integers n by the equation
P
ai
r
 n 
n
      ,
 P  i 1  pi 
 n 
where   is the Legendre symbol.
 pi 
n
We also define    1 .
1

n
If the congruence x 2  n (mod P) has a solution, we have    1 . However,
P
n
the converse is not true since   can be 1 if an even number of factors –1
P
appears.

If P and Q are odd positive integers, we have
1.
2.

 mn   m   n 

    
 P   P P
 n  n n

      
 PQ   P   Q 
3.
m  n 
     whenever m  n (mod P)
 P P
4.
 a2n   n 

    whenever (a, P )  1
 P  P
If P is an odd positive integer we have
 1
   (1)
 P
P 1
2
and
2
   (1)
P
P 2 1
8
Page 3 of 4

(Reciprocity law for Jacobi symbol) If P and Q are positive odd integers with
( P, Q)  1 , then
 P Q
    
Q  P
except when P  Q  3 (mod 4) , in which case
P
Q
    
P
Q
( P 1)( Q 1)
 P Q
Equivalently,       (1) 4 .
Q  P
P
Remark: for integer P and positive odd integer Q, ( P, Q)  1     0 .
Q
Application to Diophantine equations
Theorem. The Diophantine equation
y 2  x3  k
has no solution if k has the form
k  (4n  1) 3  4m 2 ,
where m and n are integers such that no prime p  1 (mod 4) divides m.
Proof. Assume there is a solution ( x, y ) , we want to obtain a contradiction.
Substitute the expression of k into the equation,
y 2  x 3  (4n  1) 3  4m 2 .
So y 2  x 3  1 (mod 4) and follows that x  1 (mod 4) .
On the other hand, denote a  4n  1 (note that a  1 (mod 4) ), and rewrite
the equation into
y 2  4m 2  x 3  a 3  ( x  a)( x 2  ax  a 2 ) .
Note that x 2  ax  a 2  (1) 2  (1)(1)  (1) 2  1 (mod 4) . Therefore some
prime p  1 (mod 4) divides x 2  ax  a 2 , and
y 2  4m 2  0 (mod p) .
  4m 2
However, 
 p
  1
     1 because p  1 (mod 4 ), a contradiction.
  p
Page 4 of 4