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.
Read more about this topic: Euler's Criterion
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