Euler's criterion

In mathematics, Euler's criterion is used in determining in number theory whether a given integer is a quadratic residue modulo a prime.

Definition

Euler's criterion states:

Let p be an odd prime and a an integer coprime to p. Then a is a quadratic residue modulo p (i.e. there exists a number k such that k² ≡ a (mod p)) if and only if

$a^{(p - 1) / 2} \equiv 1 \pmod p.$

As a corollary of the theorem one gets:

If a is not a square (also called a quadratic non-residue) modulo p then

$a^{(p - 1)/2} \equiv -1 \pmod p.$

Euler's criterion can be concisely reformulated using the Legendre symbol:

$\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p.$

Proof of Euler's criterion

Every number either is or isn't a quadratic residue (mod p). Also, since the square-roots of 1 are 1 and −1 (mod p), and since a^p−1 ≡ 1 (mod p) (Fermat's little theorem), a^(p−1)/2 is either 1 or −1 (mod p). This immediately implies that the criterion are equivalent to the biimplication:

a is a quadratic residue modulo p if and only if a^(p−1)/2 ≡ 1 (mod p).

We prove each direction separately.

(1) Assume a is a quadratic residue modulo p. We pick k such that k² ≡ a (mod p). Then

a^(p−1)/2 ≡ k^p−1 ≡ 1 (mod p).

The first step is valid since a ≡ b (mod n) implies a^m ≡ b^m(mod n) (see modular arithmetic#Congruence relation), while the second step is Fermat's little theorem again.

(2) Assume a^{(p − 1)/2} ≡ 1 (mod p). Then let α be a primitive root modulo p, so that a can be written as αⁱ for some i. In particular, α^i(p−1)/2 ≡ 1 (mod p). Since α is a primitive root, its multiplicative order is p − 1, so p − 1 must divide i(p − 1)/2, so i must be even. Let k ≡ α^i/2 (mod p). We finally have k² = αⁱ ≡ a (mod p).

Examples

Example 1: Finding primes for which a is a residue

Let a = 17. For which primes p is 17 a quadratic residue?

We can test prime p's manually given the formula above.

In one case, testing p = 3, we have 17^{(3 − 1)/2} = 17¹ ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.

In another case, testing p = 13, we have 17^{(13 − 1)/2} = 17⁶ ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 2² = 4.

We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.

If we keep calculating the values, we find:

(17/p) = +1 for p = {13, 19, ...} (17 is a quadratic residue modulo these values)

(17/p) = −1 for p = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).

Example 2: Finding residues given a prime modulus p

Which numbers are squares modulo 17 (quadratic residues modulo 17)?

We can manually calculate:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25 ≡ 8 (mod 17)

6² = 36 ≡ 2 (mod 17)

7² = 49 ≡ 15 (mod 17)

8² = 64 ≡ 13 (mod 17).

So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 9² ≡ (−8)² = 64 ≡ 13 (mod 17)).

We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2^{(17 − 1)/2} = 2⁸ ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3^{(17 − 1)/2} = 3⁸ ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.

Euler's criterion is related to the Law of quadratic reciprocity and is used in a definition of Euler–Jacobi pseudoprimes.