In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
Contents
Let p be an odd prime and a an integer coprime to p. Then
Euler's criterion can be concisely reformulated using the Legendre symbol:
The criterion first appeared in a 1748 paper by Euler.
Proof
The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. The fact that there are (p − 1)/2 quadratic residues and the same number of nonresidues (mod p) is proved in the article quadratic residue.
If a is 0 mod p, then
Otherwise, Fermat's little theorem says that
which can be written as
Since the integers mod p form a field, one or the other of these factors must be congruent to zero.
Now if a is a quadratic residue, a ≡ x2,
So every quadratic residue (mod p) makes the first factor zero.
Lagrange's theorem says that there can be no more than (p - 1)/2 values of a that make the first factor zero. But it is known that there are (p - 1)/2 distinct quadratic residues (mod p) (besides 0). Therefore they are precisely the residue classes that make the first factor zero. The other (p - 1)/2 residue classes, the nonresidues, must make the second factor zero, or they would not satisfy Fermat's little theorem. This is Euler's criterion.
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 it as:
12 = 122 = 432 = 942 = 1652 = 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.