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| In [[number theory]] '''Euler's criterion''' is a formula for determining whether an [[integer]] is a [[quadratic residue]] [[modular arithmetic|modulo]] a [[prime number|prime]]. Precisely,
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| Let ''p'' be an [[odd number|odd]] prime and ''a'' an integer [[coprime]] to ''p''. Then<ref>Gauss, DA, Art. 106</ref>
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| :<math>
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| a^{\tfrac{p-1}{2}} \equiv
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| \begin{cases}
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| \;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p}\\
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| -1\pmod{p}& \text{ if there is no such integer.}
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| \end{cases}
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| </math>
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| Euler's criterion can be concisely reformulated using the [[Legendre symbol]]:<ref>Hardy & Wright, thm. 83</ref>
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| :<math>
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| \left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p.
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| </math>
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| The criterion first appeared in a 1748 paper by [[Leonhard Euler|Euler]].<ref>Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive</ref>
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| ==Proof==
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| The proof uses fact that the residue classes modulo a prime number are a [[finite field|field]]. See the article [[Characteristic_(algebra)#Case_of_fields|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]].
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| [[Fermat's little theorem]] says that
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| :<math>
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| a^{p-1}\equiv 1 \pmod p
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| </math>
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| (Assume throughout this solution that a is not 0 mod p). This can be written as
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| :<math>
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| (a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0 \pmod p.
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| </math>
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| Since the integers mod ''p'' form
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| a field, one or the other of these factors must be congruent to zero.
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| Now if ''a'' is a quadratic residue, ''a'' ≡ ''x''<sup>2</sup>,
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| :<math>
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| a^{\tfrac{p-1}{2}}\equiv{x^2}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv1\pmod p.
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| </math>
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| So every quadratic residue (mod ''p'') makes the first factor zero.
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| [[Lagrange's theorem (number theory)|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. | |
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| ==Examples==
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| '''Example 1: Finding primes for which ''a'' is a residue'''
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| Let ''a'' = 17. For which primes ''p'' is 17 a quadratic residue?
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| We can test prime ''p'''s manually given the formula above.
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| In one case, testing ''p'' = 3, we have 17<sup>(3 − 1)/2</sup> = 17<sup>1</sup> ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.
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| In another case, testing ''p'' = 13, we have 17<sup>(13 − 1)/2</sup> = 17<sup>6</sup> ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 2<sup>2</sup> = 4.
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| We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.
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| If we keep calculating the values, we find:
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| :(17/''p'') = +1 for ''p'' = {13, 19, ...} (17 is a quadratic residue modulo these values)
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| :(17/''p'') = −1 for ''p'' = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).
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| '''Example 2: Finding residues given a prime modulus ''p'' '''
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| Which numbers are squares modulo 17 (quadratic residues modulo 17)?
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| We can manually calculate:
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| : 1<sup>2</sup> = 1
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| : 2<sup>2</sup> = 4
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| : 3<sup>2</sup> = 9
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| : 4<sup>2</sup> = 16
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| : 5<sup>2</sup> = 25 ≡ 8 (mod 17)
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| : 6<sup>2</sup> = 36 ≡ 2 (mod 17)
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| : 7<sup>2</sup> = 49 ≡ 15 (mod 17)
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| : 8<sup>2</sup> = 64 ≡ 13 (mod 17).
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| 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 9<sup>2</sup> ≡ (−8)<sup>2</sup> = 64 ≡ 13 (mod 17)).
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| 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<sup>(17 − 1)/2</sup> = 2<sup>8</sup> ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3<sup>(17 − 1)/2</sup> = 3<sup>8</sup> ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.
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| Euler's criterion is related to the [[Quadratic reciprocity|Law of quadratic reciprocity]] and is used in a definition of [[Euler–Jacobi pseudoprime]]s.
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| ==Notes==
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| {{reflist}}
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| ==References==
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| 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.
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Clarke | first2 = Arthur A. (translator into English)
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| | title = Disquisitiones Arithemeticae (Second, corrected edition)
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = New York
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| | year = 1986
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| | isbn = 0-387-96254-9}}
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | year = 1965
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| | isbn = 0-8284-0191-8}}
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| *{{citation
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| | last1 = Hardy | first1 = G. H.
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| | last2 = Wright | first2 = E. M.
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| | title = An Introduction to the Theory of Numbers (Fifth edition)
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| | publisher = [[Oxford University Press]]
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| | location = Oxford
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| | year = 1980
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| | isbn = 978-0-19-853171-5}}
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| *{{citation
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| | last1 = Lemmermeyer | first1 = Franz
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| | title = Reciprocity Laws: from Euler to Eisenstein
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = Berlin
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| | year = 2000
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| | isbn = 3-540-66957-4}}
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| ==External links==
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| *[http://www.math.dartmouth.edu/~euler/index.html The Euler Archive]
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| {{DEFAULTSORT:Euler's Criterion}}
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| [[Category:Modular arithmetic]]
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| [[Category:Articles containing proofs]]
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| [[Category:Quadratic residue]]
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| [[Category:Theorems about prime numbers]]
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