Criterion of Euler

In Mathématiques and more precisely in modular Arithmétique, the criterion of Euler is used in Théorie of the numbers to determine if a whole given is a quadratic Résidu modulo a Prime number.

Definition

Let us consider a prime number odd p and an entirety has first with p ; then the criterion of Euler is stated in the following way:

if has is a quadratic residue modulo p (i.e. if there exists an entirety K such as K ² ≡ has (MOD p )), then $a^ {\ frac {p-1} {2}} \ equiv 1 \ pmod p$
if has is not a quadratic residue modulo p then $a^ {\ frac {p-1} {2}} \ equiv -1 \ pmod p$ .

what is also written, by using the Symbole of Legendre, in the following way:

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

Demonstration of the criterion of Euler

On the one hand, we consider that has is a quadratic residue modulo p . We find K such as K ² ≡ has (MOD p ). Then has ^{( p − 1) /2} = K ^{p − 1} ≡ 1 (MOD p ) by the Petit theorem of Fermat.

In addition, we consider that has ^{( p − 1) /2} ≡ 1 (MOD p ). Then, is α a primitive element modulo p , in other words has = α^I. Therefore, α^{I ( p − 1) /2} ≡ 1 (MOD p ). By the small theorem of Fermat, ( p − 1) I ( p − 1) divides /2, therefore I must be even. That is to say K ≡ α^{I /2} (MOD p ). We have finally K ² = α^I ≡ has (MOD p ).

Examples

Example 1: To find the prime numbers for which has is a residue

Either has = 17. For which prime numbers p , 17 is a quadratic residue?

We can test the prime numbers p' given manually by the preceding formula.

In a case, p = 3, we test have 17^{(3 − 1) /2} = 17¹ ≡ 2 (MOD 3) ≡ -1 (MOD 3), consequently 17 is not a quadratic residue modulo 3.

In another case, p = 13, we test have 17^{(13 − 1) /2} = 17⁶ ≡ 1 (MOD 13), consequently 17 is a quadratic residue modulo 13. Like confirmation, let us note that 17 ≡ 4 (MOD 13), and 2² = 4.

We can more quickly make these calculations by using the arithmetic modular one and the properties of the symbol of Legendre.

If we calculate 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: To find the residues with a module first p given.

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

We can calculate manually:

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)

Therefore, the whole of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we do not need to calculate the squares for the values from 9 to 16, as they all are negative compared to the values square the preceding one (c.a.d. 9 ≡ −8 (MOD 17), so 9² = (−8) ² = 64 ≡ 13 (MOD 17)).

We can find the residues quadratic or check them by using the preceding formula. To test if 2 is a quadratic residue modulo 17, we calculate 2^{(17 − 1) /2} = 2⁸ ≡ 1 (MOD 17), therefore 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) thus, it is not a quadratic residue.

The Critère of Euler is connected to the quadratic Loi of reciprocity and is used in the definition of the pseudo-first numbers of Euler-Jacobi.

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