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{{distinguish|Provable prime}}
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In [[number theory]], a '''probable prime (PRP)''' is an [[integer]] that satisfies a specific condition that is satisfied by all [[prime numbers]], but which is not satisfied by most [[composite number]]s. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called [[pseudoprime]]s), the condition is generally chosen in order to make such exceptions rare.
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Fermat's test for compositeness, which is based on [[Fermat's little theorem]], works as follows: given an integer ''n'', choose some integer ''a'' [[coprime]] to ''n'' and calculate ''a''<sup>''n'' &minus; 1</sup> [[modular arithmetic|modulo]] ''n''. If the result is different from 1, then ''n'' is composite. If the result is 1, then ''n'' is likely to be prime; ''n'' is then called a '''probable prime to base''' ''a''.  A '''weak probable prime to base''' ''a'' is an integer that is a probable prime to base ''a'', but which is not a strong probable prime to base ''a'' (see below).
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A probable prime need not be prime. For example, 341 is a probable prime base 2 because 2<sup>340</sup> (mod 341) is 1, even though 341 is composite: 341 = 11·31. Therefore, 341 is, more precisely, a '''pseudoprime''' base 2.
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For a fixed base ''a'', it is unusual for a composite number to be a probable prime (that is, a pseudoprime) to that base. For example, there are only 21853 pseudoprimes base 2 that are less than 25·10<sup>9</sup> (see page 1005 of <ref name="PSW">{{cite journal|coauthors = [[John L. Selfridge]], [[Samuel S. Wagstaff, Jr.]]|title=The pseudoprimes to 25·10<sup>9</sup>|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003-1026|url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf|author = [[Carl Pomerance]]| doi=10.1090/S0025-5718-1980-0572872-7 }}</ref>).
 
== Properties ==
 
Probable primality is a basis for efficient [[primality testing]] [[algorithm]]s, which find application in [[cryptography]]. These algorithms are usually [[randomized algorithm|probabilistic]] in nature. The idea is that while there are composite probable primes to base ''a'' for any fixed ''a'', we may hope there exists some fixed ''P''&lt;1 such that for ''any'' given composite ''n'', if we choose ''a'' randomly the probability that ''n'' is pseudoprime to base ''a'' is at most ''P''.  If we repeat this test ''k'' times, choosing a new ''a'' each time, the probability of ''n'' being pseudoprime to all the ''a''s tested is hence at most ''P<sup>k</sup>'', and as this decreases exponentially, only moderate ''k'' is required to make this probability negligibly small (compared to, for example, the probability of computer hardware error).
 
This is unfortunately false for weak probable primes, because there exist [[Carmichael number]]s; but it is true for more refined notions of probable primality, such as strong probable primes (''P''&nbsp;=&nbsp;1/4, [[Miller&ndash;Rabin primality test|Miller&ndash;Rabin algorithm]]), or
Euler probable primes (''P''&nbsp;=&nbsp;1/2, [[Solovay–Strassen primality test|Solovay&ndash;Strassen algorithm]]).
 
Even when a deterministic primality proof is required, a useful first step is to test for probable primality. This can quickly eliminate (with certainty) most composites.
 
A PRP test is sometimes combined with a table of small pseudoprimes to quickly establish the primality of a given number smaller than some threshold.
 
== Variations ==
 
An '''Euler probable prime to base''' ''a'' is an integer that is indicated prime by the somewhat stronger theorem that for any prime ''p'', ''a''<sup>(''p''&nbsp;&minus;&nbsp;1)/2</sup> equals <math>(\tfrac{a}{p})</math> modulo&nbsp;''p'', where <math>(\tfrac{a}{p})</math> is the [[Legendre symbol]]. An Euler probable prime which is composite is called an [[Euler&ndash;Jacobi pseudoprime]] to base&nbsp;''a''.  The smallest Euler-Jacobi pseudoprime to base 2 is 561 (see page 1004 of <ref name="PSW"/>). There are 11347 Euler-Jacobi pseudoprimes base 2 that are less than 25·10<sup>9</sup> (page 1005 of <ref name="PSW"/>).
 
This test may be improved by using the fact that the only square roots of 1 modulo a prime are 1 and &minus;1. Write ''n''&nbsp;=&nbsp;''d''&nbsp;·&nbsp;2<sup>''s''</sup>&nbsp;+&nbsp;1, where ''d'' is odd. The number ''n'' is a '''strong probable prime (SPRP) to base''' ''a'' if one of the following conditions holds:
 
: <math>a^d\equiv 1\pmod n,\;</math>
 
: <math>a^{d\cdot 2^r}\equiv -1\pmod n\text{ for some }0\leq r\leq s-1. \, </math>
 
A composite strong probable prime to base ''a'' is called a [[strong pseudoprime]] to base ''a''. Every strong probable prime to base ''a'' is also an Euler probable prime to the same base, but not vice versa.
 
The smallest strong pseudoprime base 2 is 2047 (page 1004 of <ref name="PSW"/>). There are 4842 strong pseudoprimes base 2 that are less than 25·10<sup>9</sup> (page 1005 of <ref name="PSW"/>).
 
There are also [[Lucas pseudoprime|Lucas probable prime]]s, which are based on [[Lucas sequence]]s.  A Lucas probable prime test can be used alone. The [[Baillie-PSW primality test]] combines a Lucas test with a strong probable prime test.
 
==See also==
 
* [[Baillie-PSW primality test]]
* [[Euler&ndash;Jacobi pseudoprime]]
* [[Carmichael number]]
* [[Miller–Rabin primality test]]
* [[Provable prime]]
 
==External links==
* [http://primes.utm.edu/glossary/page.php?sort=PRP The prime glossary &ndash; Probable prime]
* [http://www.primenumbers.net/prptop/ The PRP Top 10000 (the largest known probable primes)]
 
== References ==
{{reflist}}
 
{{Prime number classes}}
 
[[Category:Pseudoprimes]]

Latest revision as of 01:03, 11 October 2014

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