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| In [[number theory]], a [[probable prime]] is a number that passes a [[primality test]].
| | 24 yr old Environmental Consultant Dominic from Amherst, really likes knotting, property developers [http://www.jmonae.com/forum/centaline-singapore-property-agency-pte-ltd-1091791 service apartments in singapore] singapore and kids. Likes to discover unknown cities and locales like Curonian Spit. |
| A '''strong probable prime''' is a number that passes a ''strong'' version of a primality test.
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| A '''strong pseudoprime''' is a [[composite number]] that passes a strong version of a primality test.
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| All primes pass these tests, but a small fraction of composites also pass, making them "[[pseudoprime|false primes]]".
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| Unlike the [[Fermat pseudoprime]]s, for which there exist numbers that are [[pseudoprimes]] to all [[coprime]] bases (the [[Carmichael numbers]]), there are no composites that are strong pseudoprimes to all bases.
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| ==Formal definition==
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| Formally, a composite number ''n'' = ''d'' · 2<sup>''s''</sup> + 1 with ''d'' being odd is called a strong (Fermat) pseudoprime to a [[relatively prime]] base ''a'' when one of the following conditions holds:
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| : <math>a^d\equiv 1\mod n</math>
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| or
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| : <math>a^{d\cdot 2^r}\equiv -1\mod n\quad\mbox{ for some }0 \leq r < s .</math>
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| (If a number ''n'' satisfies one of the above conditions and we don't yet know whether it is prime, it is more precise to refer to it as a strong [[probable prime]] to base ''a''. But if we know that ''n'' is not prime, then one may use the term strong pseudoprime.)
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| The definition of a strong pseudoprime depends on the base used; different bases have different strong pseudoprimes. The definition is trivially met if {{math|<var>a</var> ≡ ±1 mod <var>n</var>}} so these trivial bases are often excluded.
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| [[Richard K. Guy|Guy]] mistakenly gives a definition with only the first condition, which is not satisfied by all primes.<ref>[[Richard K. Guy|Guy]], ''Pseudoprimes. Euler Pseudoprimes. Strong Pseudoprimes.'' §A12 in ''Unsolved Problems in Number Theory'', 2nd ed. New York: Springer-Verlag, pp. 27-30, 1994.</ref>
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| ==Properties of strong pseudoprimes==
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| A strong pseudoprime to base ''a'' is always an [[Euler-Jacobi pseudoprime]], an [[Euler pseudoprime]] <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> and a [[Fermat pseudoprime]] to that base, but not all Euler and Fermat pseudoprimes are strong pseudoprimes. [[Carmichael number]]s may be strong pseudoprimes to some bases—for example, 561 is a strong pseudoprime to base 50—but not to all bases.
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| A composite number ''n'' is a strong pseudoprime to at most one quarter of all bases below ''n'';<ref>[[Louis Monier|Monier]], ''Evaluation and Comparison of Two Efficient Probabilistic Primality Testing Algorithms.'' ''Theoretical Computer Science'', 12 pp. 97-108, 1980.</ref><ref>[[Michael O. Rabin|Rabin]], ''Probabilistic Algorithm for Testing Primality.'' ''Journal of Number Theory'', 12 pp. 128-138, 1980.</ref> thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases. Thus given a random base, the probability that a number is a strong pseudoprime to that base is less than 1/4, forming the basis of the widely used [[Miller-Rabin primality test]].
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| However, Arnault
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| <ref name="Arnault397Digit">{{cite journal|title=Constructing Carmichael Numbers Which Are Strong Pseudoprimes to Several Bases
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| |journal=Journal of Symbolic Computation|date=August 1995|volume=20|issue=2|pages=151–161
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| |author=F. Arnault|url=http://www.sciencedirect.com/science/article/pii/S0747717185710425|doi=10.1006/jsco.1995.1042}}</ref>
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| gives a 397-digit composite number that is a strong pseudoprime to ''every'' base less than 307.
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| One way to prevent such a number from wrongfully being declared probably prime is to combine a strong probable prime test with a [[Lucas pseudoprime|Lucas probable prime]] test, as in the [[Baillie-PSW primality test]].
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| There are infinitely many strong pseudoprimes to any base.<ref name="PSW"/>
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| ==Examples==
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| The first strong pseudoprimes to base 2 are
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| :2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751, ... {{OEIS|id=A001262}}.
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| The first to base 3 are
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| :121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531, 18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139, 74593, 79003, 82513, 87913, 88573, 97567, ... {{OEIS|id=A020229}}.
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| The first to base 5 are
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| :781, 1541, 5461, 5611, 7813, 13021, 14981, 15751, 24211, 25351, 29539, 38081, 40501, 44801, 53971, 79381, ... {{OEIS|id=A020231}}.
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| By testing the above conditions to several bases, one gets somewhat more powerful primality tests than by using one base alone.
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| For example, there are only 13 numbers less than 25·10<sup>9</sup> that are strong pseudoprimes to bases 2, 3, and 5 simultaneously.
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| They are listed in Table 7 of.<ref name="PSW"/> The smallest such number is 25326001.
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| This means that, if ''n'' is less than 25326001 and ''n'' is a strong probable prime to bases 2, 3, and 5, then ''n'' is prime.
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| Carrying this further, 3825123056546413051 is the smallest number that is a strong pseudoprime to the 9 bases 2, 3, 5, 7, 11, 13, 17, 19, and 23.
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| See
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| <ref name="spspii">
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| {{cite journal|coauthors=Min Tang|title=Finding Strong Pseudoprimes to Several Bases. II
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| |journal=Mathematics of Computation|year=2003|volume=72|issue=244|pages=2085–2097
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| |author=Zhenxiang Zhang| doi=10.1090/S0025-5718-03-01545-X }}</ref>
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| and
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| .<ref name="psp9">
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| {{cite arXiv |last1=Jiang |first1=Yupeng |last2=Deng |first2=Yingpu |eprint=1207.0063
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| |title=Strong pseudoprimes to the first 9 prime bases |class=math.NT |year=2012 |version=v1 }}
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| </ref>
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| So, if ''n'' is less than 3825123056546413051 and ''n'' is a strong probable prime to these 9 bases, then ''n'' is prime.
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| == References ==
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| {{reflist}}
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| {{Classes of natural numbers}}
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| [[Category:Pseudoprimes]]
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24 yr old Environmental Consultant Dominic from Amherst, really likes knotting, property developers service apartments in singapore singapore and kids. Likes to discover unknown cities and locales like Curonian Spit.