Euler's criterion

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In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,

Let p be an odd prime and a an integer coprime to p. Then[1]


a^{\tfrac{p-1}{2}} \equiv
\begin{cases}
\;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p}\\
     -1\pmod{p}& \text{ if there is no such integer.}
\end{cases}

Euler's criterion can be concisely reformulated using the Legendre symbol:[2]


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

The criterion first appeared in a 1748 paper by Euler.[3]

Proof[edit]

The proof uses 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.

Fermat's little theorem says that


a^{p-1}\equiv 1 \pmod p

(Assume throughout this solution that a is not 0 mod p). This can be written as


(a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0 \pmod p.

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, ax2,


a^{\tfrac{p-1}{2}}\equiv{x^2}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv1\pmod p.

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 be the ones making the second factor zero. This is Euler's criterion.

Examples[edit]

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.

Notes[edit]

  1. ^ Gauss, DA, Art. 106
  2. ^ Hardy & Wright, thm. 83
  3. ^ Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive

References[edit]

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

  • Gauss, Carl Friedrich; Clarke, Arthur A. (translator into English) (1986), Disquisitiones Arithemeticae (Second, corrected edition), New York: Springer, ISBN 0-387-96254-9 
  • Gauss, Carl Friedrich; Maser, H. (translator into German) (1965), Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea, ISBN 0-8284-0191-8 

External links[edit]