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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, | |||
Let ''p'' be an [[odd number|odd]] prime and ''a'' an integer [[coprime]] to ''p''. Then<ref>Gauss, DA, Art. 106</ref> | |||
:<math> | |||
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} | |||
</math> | |||
Euler's criterion can be concisely reformulated using the [[Legendre symbol]]:<ref>Hardy & Wright, thm. 83</ref> | |||
:<math> | |||
\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p. | |||
</math> | |||
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> | |||
==Proof== | |||
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]]. | |||
[[Fermat's little theorem]] says that | |||
:<math> | |||
a^{p-1}\equiv 1 \pmod p | |||
</math> | |||
(Assume throughout this solution that a is not 0 mod p). This can be written as | |||
:<math> | |||
(a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0 \pmod p. | |||
</math> | |||
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'' ≡ ''x''<sup>2</sup>, | |||
:<math> | |||
a^{\tfrac{p-1}{2}}\equiv{x^2}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv1\pmod p. | |||
</math> | |||
So every quadratic residue (mod ''p'') makes the first factor zero. | |||
[[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. | |||
==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<sup>(3 − 1)/2</sup> = 17<sup>1</sup> ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3. | |||
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. | |||
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: | |||
: 1<sup>2</sup> = 1 | |||
: 2<sup>2</sup> = 4 | |||
: 3<sup>2</sup> = 9 | |||
: 4<sup>2</sup> = 16 | |||
: 5<sup>2</sup> = 25 ≡ 8 (mod 17) | |||
: 6<sup>2</sup> = 36 ≡ 2 (mod 17) | |||
: 7<sup>2</sup> = 49 ≡ 15 (mod 17) | |||
: 8<sup>2</sup> = 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 9<sup>2</sup> ≡ (−8)<sup>2</sup> = 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<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. | |||
Euler's criterion is related to the [[Quadratic reciprocity|Law of quadratic reciprocity]] and is used in a definition of [[Euler–Jacobi pseudoprime]]s. | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
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. | |||
*{{citation | |||
| last1 = Gauss | first1 = Carl Friedrich | |||
| last2 = Clarke | first2 = Arthur A. (translator into English) | |||
| title = Disquisitiones Arithemeticae (Second, corrected edition) | |||
| publisher = [[Springer Science+Business Media|Springer]] | |||
| location = New York | |||
| year = 1986 | |||
| isbn = 0-387-96254-9}} | |||
*{{citation | |||
| last1 = Gauss | first1 = Carl Friedrich | |||
| last2 = Maser | first2 = H. (translator into German) | |||
| title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition) | |||
| publisher = Chelsea | |||
| location = New York | |||
| year = 1965 | |||
| isbn = 0-8284-0191-8}} | |||
*{{citation | |||
| last1 = Hardy | first1 = G. H. | |||
| last2 = Wright | first2 = E. M. | |||
| title = An Introduction to the Theory of Numbers (Fifth edition) | |||
| publisher = [[Oxford University Press]] | |||
| location = Oxford | |||
| year = 1980 | |||
| isbn = 978-0-19-853171-5}} | |||
*{{citation | |||
| last1 = Lemmermeyer | first1 = Franz | |||
| title = Reciprocity Laws: from Euler to Eisenstein | |||
| publisher = [[Springer Science+Business Media|Springer]] | |||
| location = Berlin | |||
| year = 2000 | |||
| isbn = 3-540-66957-4}} | |||
==External links== | |||
*[http://www.math.dartmouth.edu/~euler/index.html The Euler Archive] | |||
{{DEFAULTSORT:Euler's Criterion}} | |||
[[Category:Modular arithmetic]] | |||
[[Category:Articles containing proofs]] | |||
[[Category:Quadratic residue]] | |||
[[Category:Theorems about prime numbers]] | |||
Revision as of 09:31, 1 February 2014
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]
Euler's criterion can be concisely reformulated using the Legendre symbol:[2]
The criterion first appeared in a 1748 paper by Euler.[3]
Proof
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
(Assume throughout this solution that a is not 0 mod p). This 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 be the ones making the second factor zero. 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:
- 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
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References
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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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010