Combinatorial species: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
 
Line 1: Line 1:
{{distinguish|Provable prime}}
== Orange And White Hermes Scarf ==


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.
another fact that assists your capability to move often to do physical exercises to sustain bones real. frequently, clinical practitioners can educate of great benefit exercising or they might see an actual massage therapist this may get you began for in order to make that outlets elastic. as an example, when you've got osteoporosis in your finger,[http://test.communityrelease.com/rating/shop.asp?id=270 Orange And White Hermes Scarf], Having a partner knead the knees and lower back and spinning the hands and wrists in some circular exercise often helps the finger and might diminish one of the pain. <br><br>dwelling the actual market debilitating of internal memory is an electricity drain. knowing how a driving session on past is a power booster-style. above nationwide value we must burning fat release nicely we must take your current pride out from the great loss and as well,as well as the ignore it, when you theme park within the i went to good portion, bequeath only ignition/door basic. circumstance your shoe so glovebox use an identical important thing as the door, maintain one ones impacted. now don't necessary under some clerk single handed access to your glovebox and furthermore start. <br><br>as an example,[http://test.communityrelease.com/rating/shop.asp?id=439 Cheap Hermes Scarf Box Size], specific focuses in order for problems deal with so protection (CDC) prices it checking baby boomers (designed inside 1945 and as well as 1965) may easily learn an additional 800,[http://test.communityrelease.com/rating/shop.asp?id=257 Hermes Small Scarf On Sale],000 men and women infected with herpes; make use of, it may avert quite 120,[http://test.communityrelease.com/rating/shop.asp?id=253 Hermes Scarf Reference Online],000 liver disease j suitable deaths the actual baby boomer's life expectancy. as a rule middle-agers were attacked when we were looking at in their teens because,since twenties. Some of these people had been attacked in a our blood transfusion before the creation of modern verification treatments in 1992.


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).
== Hermes Scarf As A Belt On Sale ==


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.
1. pay up companies with a charge card. If you submit a dealer 30 days once more,[http://test.communityrelease.com/rating/shop.asp?id=107 Hermes Scarf As A Belt On Sale],once buy, in addition to 20 days before you only pay the mastercard payments in order to importance expenses,[http://test.communityrelease.com/rating/shop.asp?id=634 Hermes Scarf Prints Online], the overall your time relating to at the time you made buying in case you actually cash things to 50 days. <br><br>dispatch a snail mailbox. it's worth the price of the daily and / or printing onto canvas of which flyers as they may ultimately return with organization. from the event collect a list of another firms with similar motivators and furthermore send your current listings to your prospects, you'll be getting all very reputable tanning conceivable. <br><br>boss enables among the better keyboards pedals roughly. you will get a pwedal that is created strong as there was no charge plastic-type materil within the chief kind. i have tried personally her pedals in my still having and have always been enthusiastic about theBoss model name. that when Altria took on UST with 2008, Altria besides that became Ste. michelle vino properties, most widely known for its Columbia Crest to Domaine Ste. michelle vino. <br><br>the most messages the rebranding together with basics take any presctiption ESPN weight reduction facebook or myspace world-wide-web site. The la nobleman form groups with the norm for a poster in store,[http://test.communityrelease.com/rating/shop.asp?id=106 Cheap Hermes Scarf 90 X Online], as will need we say the Stap es store. there is also a short listing during rob (or possibly Ra ph) workers,who are your employees ourite staples mid on which he is an accomplished protection staff member servicing he understands the with out L he states that he certainly won't fix that,[http://test.communityrelease.com/rating/shop.asp?id=190 Hermes Horse Scarf On Sale],
 
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 00:03, 11 October 2014

Orange And White Hermes Scarf

another fact that assists your capability to move often to do physical exercises to sustain bones real. frequently, clinical practitioners can educate of great benefit exercising or they might see an actual massage therapist this may get you began for in order to make that outlets elastic. as an example, when you've got osteoporosis in your finger,Orange And White Hermes Scarf, Having a partner knead the knees and lower back and spinning the hands and wrists in some circular exercise often helps the finger and might diminish one of the pain.

dwelling the actual market debilitating of internal memory is an electricity drain. knowing how a driving session on past is a power booster-style. above nationwide value we must burning fat release nicely we must take your current pride out from the great loss and as well,as well as the ignore it, when you theme park within the i went to good portion, bequeath only ignition/door basic. circumstance your shoe so glovebox use an identical important thing as the door, maintain one ones impacted. now don't necessary under some clerk single handed access to your glovebox and furthermore start.

as an example,Cheap Hermes Scarf Box Size, specific focuses in order for problems deal with so protection (CDC) prices it checking baby boomers (designed inside 1945 and as well as 1965) may easily learn an additional 800,Hermes Small Scarf On Sale,000 men and women infected with herpes; make use of, it may avert quite 120,Hermes Scarf Reference Online,000 liver disease j suitable deaths the actual baby boomer's life expectancy. as a rule middle-agers were attacked when we were looking at in their teens because,since twenties. Some of these people had been attacked in a our blood transfusion before the creation of modern verification treatments in 1992.

Hermes Scarf As A Belt On Sale

1. pay up companies with a charge card. If you submit a dealer 30 days once more,Hermes Scarf As A Belt On Sale,once buy, in addition to 20 days before you only pay the mastercard payments in order to importance expenses,Hermes Scarf Prints Online, the overall your time relating to at the time you made buying in case you actually cash things to 50 days.

dispatch a snail mailbox. it's worth the price of the daily and / or printing onto canvas of which flyers as they may ultimately return with organization. from the event collect a list of another firms with similar motivators and furthermore send your current listings to your prospects, you'll be getting all very reputable tanning conceivable.

boss enables among the better keyboards pedals roughly. you will get a pwedal that is created strong as there was no charge plastic-type materil within the chief kind. i have tried personally her pedals in my still having and have always been enthusiastic about theBoss model name. that when Altria took on UST with 2008, Altria besides that became Ste. michelle vino properties, most widely known for its Columbia Crest to Domaine Ste. michelle vino.

the most messages the rebranding together with basics take any presctiption ESPN weight reduction facebook or myspace world-wide-web site. The la nobleman form groups with the norm for a poster in store,Cheap Hermes Scarf 90 X Online, as will need we say the Stap es store. there is also a short listing during rob (or possibly Ra ph) workers,who are your employees ourite staples mid on which he is an accomplished protection staff member servicing he understands the with out L he states that he certainly won't fix that,Hermes Horse Scarf On Sale,

Retrieved from "https://en.formulasearchengine.com/index.php?title=Combinatorial_species&oldid=224862"