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Topic: Strong pseudoprime

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	NationMaster - Encyclopedia: Strong pseudoprime
		A strong pseudoprime to base a is always an Euler pseudoprime to base a (Pomerance, Selfridge, Wagstaff 1980), but not all Euler pseudoprimes are strong pseudoprimes.
		As Monier and Rabin showed in 1980, a composite number n is a strong pseudoprime to at most one quarter of all bases "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases.
		A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number.
	www.nationmaster.com /encyclopedia/Strong-pseudoprime (604 words)

	Pseudoprime - Wikipedia, the free encyclopedia
		A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime.
		A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number.
		Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (sequence A001567 in OEIS).
	en.wikipedia.org /wiki/Pseudoprime (418 words)

	Pseudoprime. Who is Pseudoprime? What is Pseudoprime? Where is Pseudoprime? Definition of Pseudoprime. Meaning of ...
		In general, an integer which has a certain property shared by all prime numbers, but is itself not prime, is called a pseudoprime for that particular property.
		The most important class of pseudoprimes come from the Fermat's little theorem and hence they are called Fermat pseudoprimes.
		Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (SIDN A001567).
	www.knowledgerush.com /kr/encyclopedia/Pseudoprime (441 words)

	moviestore.ca - Strong pseudoprime (Site not responding. Last check: 2007-10-10)
		The Smallest Strong Pseudoprime (base-2) which is a multiple of 3 is 5455590801.
		A strong pseudoprime to base a is always an Euler pseudoprime to base a (Pomerance, Selfridge, Wagstaff 1980), but not all Euler pseudoprime s are strong pseudoprimes.
		Some Fermat pseudoprime s and Carmichael number s are also strong pseudoprimes.
	www.moviestore.ca /Strong-pseudoprime/reference/fullview/wikipedia/1100422 (259 words)

	PlanetMath: Perrin pseudoprime
		The first few Perrin pseudoprimes are 271441, 904631, 16532714, 24658561, 27422714, 27664033, 46672291 (listed in A013998 of Sloane's OEIS).
		Unlike Perrin pseudoprimes, most other pseudoprimes are pseudoprimes because of a congruence relation to a given base.
		This is version 2 of Perrin pseudoprime, born on 2006-08-23, modified 2007-03-27.
	www.planetmath.org /encyclopedia/PerrinPseudoprime.html (132 words)

	Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana
		Strong pseudoprimes to base a are less common than Euler pseudoprimes.
		We may observe that 91 is thus coprime to twice as many bases as it's a pseudoprime to (72 is the Euler totient of 91).
		An integer n may not be a strong pseudoprime to more than ¼ of the possible bases.
	home.att.net /~numericana/answer/pseudo.htm (3004 words)

	GNU GMP mpz_probab_prime_p Pseudoprimes
		Such an integer will also be referred to as an mpz_spsp of order k, or simply an mpz_spspk---a strong pseudoprime to k bases for the function mpz_probab_prime_p---or generically as an mpz_spsp.
		It is characterized by the fact that the performance of k repetitions of Miller's test by mpz_probab_prime_p is insufficient to resolve N as a composite, whereas k+1 repetitions is sufficient.
		However, the global frequency of pseudoprimes is ultimately a consequence of the number of bases k, so the global frequency has not changed significantly.
	www.trnicely.net /misc/mpzspsp.html (1637 words)

	Pseudoprime (Site not responding. Last check: 2007-10-10)
		The most important class of pseudoprime s come from the Fermat's little theorem and hence they are called Fermat pseudoprime s.
		Pseudoprime s to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (SIDN A001567).
		The first smallest pseudoprime s for bases a ≤ 200 are given in the following table; the colors mark the number of prime factors.
	www.portaljuice.com /pseudoprime.html (441 words)

	Clearing up the market cycle... best Pseudoprime (Site not responding. Last check: 2007-10-10)
		A strong pseudoprime to a base is an Odd Composite Number with (for Odd) for which either...
		Pseudoprim e -- from MathWorld Pseudoprime -- from MathWorld A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers.
		A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which...
	ascot.pl /th/Fourier5/Pseudoprime.htm (630 words)

	Number Theory Glossary
		Also called the strong pseudoprime test, this test was originally proposed by M. Rabin in Algorithms and Complexity, J. Traub (Ed), Academic Press, 1976, pp 35-36., based on ideas of G. Miller.
		Also called the Euler pseudoprime test, this test was originally proposed by Solovay and Strassen in SIAM J. Computing, 6 (1977), 84-85 and 7 (1978), 118.
		If an integer is a strong pseudoprime it is also a Fermat pseudoprime and an Euler pseudoprime.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (828 words)

	[No title]
		One reason that pseudoprimes based on recurrence sequences have attracted interest is that the pseudoprimes for these sequence are often different from ordinary pseudoprimes.
		We introduce the concept of Frobenius pseudoprimes, not to inflict a new and different notion of pseudoprimality on the mathematical world, but to show that many existing pseudoprimality tests can be generalized and described in terms of finite fields.
		A {\bf Frobenius pseudoprime} with respect to a monic polynomial $f(x)$ is a composite which is a Frobenius probable prime with respect to $f(x)$.
	www.pseudoprime.com /pseudo1.tex (5982 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-10)
		It's a little slower then finding out whether M is a strong pseudoprime to every base, which is what you'd really have to do to really show that your number is really a prime, and not just some imposter.
		If you find that M is a strong pseudoprime to the base 2 and M is smaller than 2047, then M is prime.
		If you test it again with the base 3, and it's again a strong pseudoprime, then M is a prime provided it's less than 1373,653.
	www.bonus.com /contour/askdrmath/http@@/mathforum.org/library/drmath/view/51512.html (690 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		If lucas(n) returns 1, then n is a strong base 2 pseudoprime and a Lucas probable prime; if lucas(n) returns 0, then n is composite.
		At the end of this paper it is conjectured that if n is a strong base 2 pseudoprime and a Lucas probable prime, then n is in fact a prime.
		At the end of this paper * it is conjectured that if n is a strong base 2 pseudoprime and a Lucas * probable prime, then n is in fact a prime.
	www.numbertheory.org /gnubc/lucas (387 words)

	Clearing up the market cycle... best Strong Pseudoprime (Site not responding. Last check: 2007-10-10)
		A number with is a strong Frobenius pseudoprime with respect to Iff is a Strong Pseudoprime with respect to...
		Stron g Pseudoprime -- from MathWorld Strong Pseudoprime -- from MathWorld A strong pseudoprime to a base a is an odd composite number n with n-1=d\cdot 2^s (for d odd) for which either a^d\equiv 1\ \left({{\rm mod\ } {n}}\right) or a^{d\...
		A strong pseudoprime to a base a is an ODD COMPOSITE Number n with n-1...
	ascot.pl /th/Fourier5/Strong-Pseudoprime.htm (530 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Title On Carmichael numbers with 3 factors and the strong pseudoprime test Author Warren D. Smith Abstract The well known ``strong pseudoprime test'' has its highest probability of error ($\approx 1/4$) when the numbers being tested are certain Carmichael numbers with 3 prime factors.
		1/8$ in the strong pseudoprime test are certain numbers with only 2 prime factors and certain prime powers.
		However if these cases are somehow known to apply, then we show how to improve the strong pseudoprime test so that its probability of error on $N$ is $O ( 1/\sqrt{\ln N})$.
	www.math.temple.edu /~wds/homepage/carm3.sue (139 words)

	[No title]
		Here are some functions which can be used to find pseudoprimes, strong pseodoprimes and implement a probabilisitic primality test.
		To find the smallest pseudoprime to base 5 which is larger than 5, we can type :[font = input; preserveAspect; startGroup; nowordwrap; ] n=5; While[ !pspQ[5,n], n++]; Print[n] :[font = print; inactive; preserveAspect; endGroup; nowordwrap; ] 124 :[font = text; inactive; preserveAspect; nohscroll; ] We verify if 124 is a pseudoprime to base 5.
		Here is a program which tests if a number n passes the strong pseudoprime test to base a.
	www.math.columbia.edu /~rama/spsp.ma (506 words)

	PlanetMath: Miller-Rabin prime test
		is called a strong pseudoprime in the basis
		Note that this theorem states that there are no such things as Carmichael numbers for strong pseudoprimes (i.e.
		composite numbers that are strong pseudoprimes for all values of
	planetmath.org /encyclopedia/StrongPseudoprime.html (209 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-10)
		Then you pick a number (call it A) that's less than M. Raise A to the power M-1, and find what it's congruent to Mod M. If it's not congruent to 1, then M isn't even a pseudoprime (which is different than a strong pseudoprime) to the base A, so it's composite.
		That first set of equal signs means "is congruent to", and the second set means "is not congruent to." Note that this gives an airtight proof of the primality of a number.
		If you test it again to the base 5, and it's again a strong pseudoprime, then M is prime provided it's less than 25,326,001.
	mathforum.org /library/drmath/view/51512.html (681 words)

	Lucas Lehmer + Rabin-Miller + ?
		The number must be a Fibonacci pseudoprime." = = = = There is no composite known to pass all of these tests according to = = the writer.
		Indeed it is claimed here: http://groups.yahoo.com/group/primenumbers/message/8548 that 252,601 is also a strong pseudoprime base 2, but "according to Crandall-Pomerance book, Baillie-PSW exclude it, because n=1 mod 5." The code and output here appears to confirm its status as a Fibonacci pseudoprime: http://www.cs.rit.edu/usr/local/pub/pga/lucas-PP This would seem to make the condition 1.
		The number must be a Fibonacci pseudoprime." There is no composite known to pass all of these tests according to the writer.
	www.forum-one.org /new-5093556-4346.html (533 words)

	Math Forum Discussions - Re: Can a strong pseudoprime be a square ?
		Re: Can a strong pseudoprime be a square ?
		Can a strong pseudoprime be a square (revisited) ?
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=86906&messageID=424648 (590 words)

	Baillie-PSW Primality Test
		Perform a Miller-Rabin (strong probable prime) test, base 2, on N. Perform a (standard or strong) Lucas-Selfridge test on N, using Lucas sequences with the parameters suggested by Selfridge.
		As of this date, both the standard and strong tests are flawless; no counterexample (Baillie-PSW standard or strong pseudoprime) is known.
		The strong Lucas-Selfridge test produces only roughly 30 % as many pseudoprimes as the standard version; for example, among the odd composites N < 10^6, there are 219 standard Lucas-Selfridge pseudoprimes, 58 strong Lucas-Selfridge pseudoprimes, and 46 base-2 strong pseudoprimes (these totals presume no screening with odd trial divisors).
	www.trnicely.net /misc/bpsw.html (957 words)

	MathGroup Archive (2000/07) - Re: PrimeQ queries (Site not responding. Last check: 2007-10-10)
		The PrimeQ function combines two strong pseudoprime tests (base 2 and base 3) and the Lucas pseudoprime test, with the results correct up to 10^16 (no known counterexamples and all known primes correctly identified).
		Chapter 2 (Prime Numbers) is more basic, but includes a review of strong pseudoprime tests, which are based on Fermat's Little theorem: if p is prime then a^(p-1) congruent to 1 (mod p) if gcd(a,p) =1.
		For the 2-strong pseudoprime test, for example, we might infer that if n is odd and 2^(n-1) is congruent to 1 (mod 1), then n is prime.
	hilbert.math.hr /arhive/mathgroup/2000/07/0037.html (339 words)

	The Prime Glossary: pseudoprime
		A probable-prime which is composite is called a pseudoprime.
		(At one time all probable primes were called pseudoprimes, but now the terminology has been corrected.) The smallest examples of pseudoprimes for bases 2, 3, 5, and 7 are as follows.
		It is harder to find examples of composites which are probable primes (or better, strong probable primes) to several bases, but this is always possible [AGP94a].
	primes.utm.edu /glossary/page.php?sort=Pseudoprime (311 words)

	C# BigInteger Class - The Code Project - C# Programming
		This is known as probabilistic primality testing and numbers that passes the test are known as pseudoprimes.
		Test whether the number is a strong pseudoprime to base 2.
		This test works based on the assumption that it is extremely rare for a composite number to be both a base 2 strong pseudoprime and a strong Lucas pseudoprime.
	www.codeproject.com /csharp/biginteger.asp (2779 words)

	Math 5410 Midterm Exam
		(b) Prove that any Euler pseudoprime to the base b is a pseudoprime to the base b.
		- 1 is a strong pseudoprime and an Euler pseudoprime to the base 2.
		This is a cyclic shift cipher by 13 (i.e.
	www-math.cudenver.edu /~wcherowi/courses/m5410/m5410mid.html (970 words)

	Strong pseudoprime
		+ 1 is called a strong pseudoprime to base a iff one of the following conditions hold:
		; the first few strong pseudoprimes to base 3 are 121, 703, 1891, 3281, 8401, 8911, 10585,...
		C. Pomerance, J. Selfridge and Wagstaff, Jr., S. The pseudoprimes to 25 · 10
	www.xasa.com /wiki/en/wikipedia/s/st/strong_pseudoprime.html (210 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-10-10)
		+ 1 is called a strong pseudoprime to base ''a'' iff one of the following conditions hold: :
		As Monier and Rabin showed in 1980, a composite number ''n'' is a strong pseudoprime to at most one quarter of all bases <''n''; thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases.
		There you find a list of all editors and the possibility to edit the original text of the article Strong pseudoprime.
	www.mauspfeil.net /Strong_pseudoprime.html (268 words)

	Amazon.com: "strong pseudoprime": Key Phrase page (Site not responding. Last check: 2007-10-10)
		But we are now equipped with the strong pseudoprime test which...
		A' is a strong pseudoprime to the base a if (12.21) af - 1 (mo(1 N) or a"r - - 1 (mod N) for some...
		Then n is called a strong pseudoprime to the base a if ad = 1 (mod n) or a2' `r - -1 (mod n) for some r,...
	www.amazon.com /phrase/strong-pseudoprime (389 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		It runs the strong pseudoprime test with the bases 2,3,5,7, so that a number passing this test with result 2 is at least a strong pseudoprime to these bases.
		MILRAB performs a strong pseudoprime test on the number N in level 2 to the base B in level 1.
		The output is 1 if N is a strong base-B pseudoprime and 0 if not.
	wwwbzs.tu-graz.ac.at /~fleischh/computer/hp48/horn/math/faktoren/prime002.txt (425 words)

	Prime Numbers:
		Using pseudoprimes (composites that appear to be primes) as
		It is known that a pseudoprime will pass this test for at most one fourth of all bases less than itself, but we can do better.
		Jaeschke (1993) showed that there are only 101 strong pseudoprimes for the bases 2, 3, and 5 less than 10
	math.arizona.edu /~ura/021/Singleton.Travis/resources/prime.htm (273 words)

	Pseudoprime tests
		The strong pseudoprime test is much better at detecting composite numbers than the traditional Fermat pseudoprime test.
		The smallest number that is a strong pseudoprime base 2 and 3 and 5 is 25326001
		that is a strong pseudoprime base 2 and 3 and 5 and 7 is 3215031751.
	www.math.uic.edu /~jeremy/math436/main/node11.html (692 words)

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