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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>
 
:<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''&nbsp;−&nbsp;1)/2 values of ''a'' that make the first factor zero. But it is known that there are (''p''&nbsp;−&nbsp;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''&nbsp;−&nbsp;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]]

Latest revision as of 19:49, 3 January 2015

I gained a great deal of weight when I was pregnant with my son, and my body happened to shop really regarding each ounce left over for the next four years. It didn't feel advantageous understanding I hadn't reduction anything, plus I'd had enough of feeling fat all time. After seeing how heavy I really looked inside my brother's marriage images, I decided to get rid of weight by setting a New Year's resolution!

Over the last 100 years the average person has gotten taller and tends to carry more muscle. The result is that BMI calculations tend to be a little bit off, many individuals might read high than they truly are. However for most persons the results are still fairly accurate. If you are surprisingly tall or you're carrying a great deal of muscle be prepared for the charts to tell we that you are overweight.

Many folks go their whole lives without ever breaking any bones. If someone suddenly develops a high likeliness for fractures due to brittle bones it can indicate osteoporosis, which in turn is a side impact of malnurition. Granted, in some cases this might be age-related. However not consuming enough calories and calcium will weaken even a young person's bones.

Knowing your BMI through the bmi calculator females female a female will get a better control over her body, weight plus looks. She may feel more confident whether or not she may not have the ideal vital statistics. A female, specifically whenever she is expecting or is feeding her baby, should ignore her BMI though. At the same time, some amount of body fat in a woman's body actually helps her reproductive system. If a woman has body fat that is lower than twenty %, may even have irregular periods and issues conceiving a child. Body fat, whenever not inside excess, assists women to fight against osteoporosis.

If you be enthusiastic to create adjustments inside the lifestyle, invest a day recording all calories we eat. What we imagine you eat might be a remarkable deal lower than what you actually eat. If you never count, you do not understand. Always be sure to accurately record calories to the proper part size. CalorieKing.com is a superb destination to look for any kind of caloric values not found on a nutritional label. They have a book variation that I have always noticed to be very handy.

The health definition of the calorie is the volume of power it usually take to improve the temperature of 1 gram of water, 1 degree Celsius. A calorie is actually a measurement of stamina. Similar to putting gas in a vehicle, calories would be the power which will make the body operate correctly. When we are not able to put enough calories inside the body, bodily organs couldn't work. When you place a lot of calories in the bodies, we gain fat.

In conclusion, DO NOT utilize BMI because an accurate gauge for the weight, you may end up in tears like Sally, trust anything more exact like body fat percentage, or conversely, lean body mass.

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