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The '''Fermat primality test''' is a [[randomized algorithm|probabilistic]] test to determine if a number is [[probable prime]].
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==Concept==
[[Fermat's little theorem]] states that if ''p'' is prime and <math>1 \le a < p</math>, then
 
:<math>a^{p-1} \equiv 1 \pmod{p}.</math>
 
If we want to test if ''p'' is prime, then we can pick random ''a'''s in the interval and see if the equality holds. If the equality does not hold for a value of ''a'', then ''p'' is composite. If the equality does hold for many values of ''a'', then we can say that ''p'' is [[probable prime]].
 
It might be in our tests that we do not pick any value for ''a'' such that the equality fails. Any ''a'' such that
:<math>a^{n-1} \equiv 1 \pmod{n}</math>
when ''n'' is composite is known as a ''Fermat liar''. Vice versa, in this case ''n'' is called [[Fermat pseudoprime]] to base ''a''.
 
If we do pick an ''a'' such that
:<math>a^{n-1} \not\equiv 1 \pmod{n}</math>
then ''a'' is known as a ''Fermat witness'' for the compositeness of ''n''.
 
== Example ==
Suppose we wish to determine if ''n''&nbsp;=&nbsp;221 is prime. Randomly pick 1 ≤ ''a'' < 221, say ''a''&nbsp;=&nbsp;38. We check the above equality and find that it holds:
:<math>a^{n-1} = 38^{220} \equiv 1 \pmod{221}.</math>
Either 221 is prime, or 38 is a Fermat liar, so we take another ''a'', say 26:
:<math>a^{n-1} = 26^{220} \equiv 169 \not\equiv 1 \pmod{221}.</math>
So 221 is composite and 38 was indeed a Fermat liar.
 
==Algorithm and running time==
The algorithm can be written as follows:
:'''Inputs''': ''n'': a value to test for primality; ''k'': a parameter that determines the number of times to test for primality
:'''Output''': ''composite'' if ''n'' is composite, otherwise ''probably prime''
:Repeat ''k'' times:
::Pick ''a'' randomly in the range [1, ''n'' &minus; 1]
::If <math>a^{n-1}\not\equiv1 \pmod n</math>, then return ''composite''
:If composite is never returned: return ''probably prime''
 
Using fast algorithms for [[modular exponentiation]], the running time of this algorithm is [[Big O notation|O]](''k'' × log<sup>2</sup>''n'' × log log ''n'' × log log log ''n''), where ''k'' is the number of times we test a random ''a'', and ''n'' is the value we want to test for primality.
 
==Flaw==
There are infinitely many values of <math>n</math> (known as [[Carmichael number]]s) for which <u>all</u> values of <math>a</math> for which <math>gcd(a, n) = 1</math> are Fermat liars. While Carmichael numbers are substantially rarer than prime numbers,<ref>Erdös' upper bound for the number of Carmichael numbers is lower than the prime number function n/log(n)</ref> there are enough of them that Fermat's primality test is often not used in the above form.  Instead, other more powerful extensions of the Fermat test, such as [[Baillie-PSW primality test|Baillie-PSW]], [[Miller-Rabin primality test|Miller-Rabin]], and [[Solovay-Strassen primality test|Solovay-Strassen]] are more commonly used.
 
In general, if <math>n</math> is not a Carmichael number then at least half of all
 
:<math>a\in(\mathbb{Z}/n\mathbb{Z})^*</math>
 
are Fermat witnesses.  For proof of this, let <math>a</math> be a Fermat witness and <math>a_1</math>, <math>a_2</math>, ..., <math>a_s</math> be Fermat liars.  Then
 
:<math>(a\cdot a_i)^{n-1} \equiv a^{n-1}\cdot a_i^{n-1} \equiv a^{n-1} \not\equiv 1\pmod{n}</math>
 
and so all <math>a \times a_i</math> for <math>i = 1, 2, ..., s</math> are Fermat witnesses.
 
==Applications==
The encryption program [[Pretty Good Privacy|PGP]] uses this primality test in its algorithms. The chance of PGP generating a [[Carmichael number]] is less than 1 in 10<sup>50</sup>, which is more than adequate for practical purposes.
 
==References==
* {{cite book |author=[[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], [[Clifford Stein]] |title=[[Introduction to Algorithms]] |edition=Second Edition |publisher=MIT Press; McGraw-Hill |year=2001 |isbn=0-262-03293-7 |page=889&ndash;890 |chapter=Section 31.8: Primality testing}}
{{reflist}}
 
{{number theoretic algorithms}}
 
[[Category:Primality tests]]
[[Category:Modular arithmetic]]

Revision as of 18:17, 18 February 2014

Are we one of the 50% - 70% of adults who suffer from hemorrhoids? If you are, you may be almost certainly here interested in several efficient hemorrhoid treatments. And like me you'd probably want a hemorrhoid natural cure, when possible.

Another kind of hemorrhoid relief is the pill which is to be swallowed. It assists to control blood flows plus pressure in the body system. This assists to tighten vein tissues plus this provides a relaxation for the hemorrhoid area. This but has some negative effects which could not augur well for several persons. Pharmacists can really not prescribe this option because of the side effects.

For example, inside the case of constipation you'd make certain that your diet associated more fiber and water and less processed food. This really is not the whole pic and is a bit too simplistic, however you get the general idea. Another benefit to utilizing natural techniques when dealing with all the irregularity matter is the fact that not just would this remedy the constipation, but there are definite total health advantages to eating correctly.

A healthy digestion will additionally be a key for not needing to undertake any hemorrhoid cure. Good digestion ehances regular bowel movement. Exercising daily, walking for regarding 20 to 30 minutes a day supports the digestive system.

So, he HAD heard about a hemorrhoids house remedy or two, he said, yet couldn't absolutely remember any details regarding them. I told him I needed time to think about the upcoming step and got out of there and into the bright sunshine as fast because I may. Surgery for hemorrhoids certainly wasn't my initial choice.

It is moreover advisable to drink enough water each day. Staying hydrated could enable anybody who has to live with all the daily pain of hemorrhoids. So aim to drink around 8 glasses of water every day plus you will be hitting the suggested amount for hydration. Not drinking enough water will cause constipation, that is among the causes of hemorrhoids.

Depending found on the severerity of the symptom, certain advertised their hemorrhoids were cure inside 48 hours. I have tried it plus they are quite safe and has no side impact as they use all-natural ingredience.

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