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	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 - Wikipedia (Site not responding. Last check: 2007-10-09)
		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) (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001567).
	wikipedia.findthelinks.com /ps/Pseudoprime.html (431 words)

	Pseudoprime -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		A pseudoprime is a (additional info and facts about probable prime) probable prime (an (Any of the natural numbers (positive or negative) or zero) integer which shares a property common to all (An integer that has no integral factors but itself and 1) prime numbers) which is not actually prime.
		The most important class of pseudoprimes come from (additional info and facts about Fermat's little theorem) Fermat's little theorem and hence are called Fermat pseudoprimes.
		A number x that is a pseudoprime for all values of a that are coprime to x is called a (additional info and facts about Carmichael number) Carmichael number.
	www.absoluteastronomy.com /encyclopedia/P/Ps/Pseudoprime.htm (624 words)

	Pseudoprime (Site not responding. Last check: 2007-10-09)
		Pseudoprimes can be classified by according to which property theysatisfy.
		A number x that is a pseudoprime for all values ofa that are coprime to x is called a Carmichaelnumber.
		Pseudoprimes to base 2 are called Pouletnumbers or sometimes Sarrus numbers or Fermatians (sequence A001567 in OEIS).
	www.therfcc.org /pseudoprime-150946.html (473 words)

	Lucas and Perrin Pseudoprimes (Site not responding. Last check: 2007-10-09)
		Lucas pseudoprimes are discussed in Ribenboim's "The Book of Prime Number Records" (Springer, 1988), along with the algebraic identities that can be used to compute the nth Lucas number in O(log(n)) steps.
		Incidentally, the table of Lucas pseudoprimes (of the second kind, corresponding to the quadratic x^2 - x - 1) on page 104 of the first edition purports to give the 25 such numbers less than 1E+5, but actually lists only 24 numbers.
		Eleven of the remaining pseudoprimes can be excluded because they have the wrong Jacobi symbol, so this leaves just 56 composites less than a million that cannot be distinguished from primes based on the quadratic x^2 - x - 1.
	www.mathpages.com /home/kmath127.htm (632 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		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)

	Symmetric Pseudoprimes
		In other words, given a monic polynomial f with integer coefficients, a symmetric pseudoprime relative to f is defined as a composite integer N such that every elementary symmetric function of the Nth powers of the roots of f is congruent (mod N) to the same function of the first powers.
		Symmetric pseudoprimes tend to be more rare relative to polynomials with larger Galois groups.
		In the articles listed below, basic propositions and computational techniques associated with symmetric pseudoprimes are presented, along with specific examples relative to selected polynomials of degrees 1 to 5, and the final article describes complete congruence conditions on the terms of arbitrary linear recurring sequences.
	mathpages.com /home/kmath003/kmath003.htm (590 words)

	Number Theory Glossary (Site not responding. Last check: 2007-10-09)
		A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases.
		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 (827 words)

	[No title]
		In fact, nobody has claimed the \$620 offered for a Lucas pseudoprime with parameters $(1,-1)$ (see \S 2 for a definition of this term), congruent to $2$ or $3\bmod 5$, that is also a pseudoprime to the base $2$ \cite{\psw, \guy}.
		A {\bf Lucas pseudoprime} with parameters $(P,Q)$ is a composite $n$ with $(n,2Q\Delta)=1$ such that $U_{n-\left(\Delta\over n\right)}\equiv 0 \bmod n$.
		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)

	[No title] (Site not responding. Last check: 2007-10-09)
		Then a symmetric pseudoprime relative to f is a composite integer c such that f(z^c)=0 (mod c).
		Essentially, it's very difficult to construct a pseudoprime out of anything except "splitting primes", i.e., primes p such that the polynomial f splits into linear factors in the field Z_p.
		For a polynomial of degree d with the fully symmetric group S_d, the proportion of all primes that are splitting primes is 1/(d!).
	www.math.niu.edu /~rusin/papers/known-math/94/primalty.tst (1695 words)

	Lucas pseudoprime -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, Lucas pseudoprimes in (additional info and facts about number theory) number theory are defined in terms of (additional info and facts about Lucas sequence) Lucas sequences.
		If p is an odd (An integer that has no integral factors but itself and 1) prime number for which the (additional info and facts about Jacobi symbol) Jacobi symbol
		These numbers are called Lucas pseudoprimes, named by analogy with (additional info and facts about pseudoprime) pseudoprimes.
	www.absoluteastronomy.com /encyclopedia/L/Lu/Lucas_pseudoprime.htm (121 words)

	Section 7
		We can easily apply trial division and then pseudoprimality tests so that we don't have to waste our time with numbers that are definitely composite, which leaves us with large probable primes that need primality to be confirmed by additional means.
		For the two types of pseudoprime already mentioned, there is no measure of the probability of composite n surviving repeated tests; we just accept that it decreases substantially at each iteration.
		It is better to use the repeat pseudoprime method for a smaller range of test values, and using the fact that the probability of a composite n passing each iteration of the test is at most 1/4, so that after k successful tests, the probability that n is prime is greater than 1
	www.glasgowg43.freeserve.co.uk /pfaq7.htm (2603 words)

	Pseudoprime Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-09)
		Looking For pseudoprime - Find pseudoprime and more at Lycos Search.
		Find pseudoprime - Your relevant result is a click away!
		Look for pseudoprime - Find pseudoprime at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Pseudoprime (556 words)

	Pseudoprime - Encyclopedia Glossary Meaning Explanation Pseudoprime (Site not responding. Last check: 2007-10-09)
		Here you will find more informations about Pseudoprime.
		Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians.
		base-2 strong pseudoprimes** (sequence A001262 in OEIS)
	www.encyclopedia-glossary.com /en/Pseudoprime.html (370 words)

	Algebra-Cryptology Seminar
		The Lucas pseudoprime test is based on a divisibility condition of the Lucas sequence, which is a second order recursive sequence.
		The problem with this method is that classical pseudoprime tests are ``dependent'' in the sense that if n is pseudoprime with respect to one basis a then it is ``likely'' that n will be pseudoprime with respect to another basis b also.
		According to a conjecture, using the Lucas pseudoprime test together with the classical pseudoprime test breaks this dependence and hence produces a very effective primality test.
	www.math.fau.edu /schmidme/ACS-fall2003.html (1068 words)

	MathGroup Archive: July 2000 [00038]
		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.
	forums.wolfram.com /mathgroup/archive/2000/Jul/msg00038.html (338 words)

	Citations: Implementation of a new primality test - Cohen, Lenstra (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		At present, one is able to prove primality of numbers with 1000 and more digits [4, 16] For an excellent treatment of old and modern primality tests, see [31, pp.
		....of the numbers on the right, then N is called a strong pseudoprime to base a (spsp a) As for psp a s, there is an infinite number of spsp a.
		Numbers which pass this test, but are nevertheless composite, are called pseudoprimes.
	citeseer.ist.psu.edu /context/319004/0 (2956 words)

	Baillie-PSW Primality Test
		Selfridge specified the following parameters for the generation of the Lucas sequences: P = 1 and Q = (1 - D)/4, where D is the first integer in the sequence {5, -7, 9, -11, 13, -15,...} for which GCD(D,N)=1 and the Jacobi symbol (DN) = -1.
		Code is also included for the "extra strong" Lucas test, as developed by Zhaiyu Mo and James P. Jones (circa 1997), and described by Jon Grantham (1998).
		However, I have not employed the extra strong Lucas test in the Baillie-PSW test, as the Lucas sequence parameters are inconsistent with those of the Lucas-Selfridge tests; consequently, the extra strong Lucas pseudoprimes were not found to be disjoint from those of the Miller-Rabin test with base 2 (or any other single Miller-Rabin base employed).
	www.trnicely.net /misc/bpsw.html (924 words)

	Reference.com/Encyclopedia/25 (number)
		25 is the smallest pseudoprime satisfying the congruence 7
		It is also a Somer-Lucas pseudoprime based on 6.
		Within base 10 one can readily test for divisibility by 25 by seeing if the last two digits of the number match 25, 50, 75 or 00.
	www.reference.com /browse/wiki/25th (401 words)

	PSEUDOPRIME
		This term is an esoteric pun derived from number theory: a number that passes a certain kind of "primality test" may be called a `pseudoprime' (all primes pass any such test, but so do some composite numbers), and any number that passes several is, in some sense, almost certainly prime.
		The hacker backgammon usage stems from the idea that a pseudoprime is almost as good as a prime: it will do the same job unless you are unlucky.
		Source: compiled by the editor from various references; see credits.
	www.websters-online-dictionary.org /ps/pseudoprime.html (724 words)

	[No title]
		Lucas is also a possible setting to the type option in PseudoprimeSearch."; LucasCertificate::usage = "LucasCertificate[p] gives a certificate of p's primality using a Lucas test in a recursive fashion.
		It requires a factorization of p+1."; LucasParameter::usage = "LucasParameter[n] finds a P value that can be paired with Q=1 to form a strong Lucas test."; LucasParameters::usage = "LucasParameters is an option to PseudoprimeSearch that specifies the parameters for the Lucas sequence.
		The determination is by the Lucas-Lehmer test."; MethodA::usage = "MethodA[n, start:5] gives the Lucas sequence parameters {1, Q} that can be used in a Lucas pseudoprime test on n.
	www.macalester.edu /~bressoud/books/CNT.m (4172 words)

	MathGroup Archive: September 2001 [00142]
		I want to know the comparison of speed of one lucas test versus speed of one strong pseudoprime test to one base say base 2.
		Certainly Lucas test is more time consuming than strong pseudoprime test.
		Will it be speedier if PrimeQ function is based on 7 Strong pseudoprime test rather than 2 strong pseudoprime test and one lucas test for prime certification of numbers upto 10^16.
	forums.wolfram.com /mathgroup/archive/2001/Sep/msg00142.html (118 words)

	Pseudoprime - Encyclopedia, History, Geography and Biography
		base-2 strong pseudoprimes (sequence A001262 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001262) in OEIS)
		This page was last modified 07:40, 2 Mar 2005.
		This encyclopedia, history, geography and biography article about Pseudoprime contains research on
	www.arikah.net /encyclopedia/Pseudoprime (461 words)

	DESCRIPTION OF CALC (Site not responding. Last check: 2007-10-09)
		This finds the first integer after x which passes the strong base 2 pseudoprime test and the Lucas pseudoprime test.
		n is subjected to a strong pseudoprime test to base 2, together with a Lucas pseudoprime test.
		If z=0 is returned, n is composite, while if z=1 is returned, then n is a Lucas probable prime, as well as a base 2 strong pseudoprime.
	www.itee.uq.edu.au /~gap/calc/krm_calc.html (2347 words)

	Pseudoprime (Site not responding. Last check: 2007-10-09)
		If a number x isn't prime, a is coprime to x and x divides a
		All is still licensed under the GNU FDL.
		But while a man can sleep, life is rendered at least there is none so precious as that same sleep, which, as wise Sancho rest, so sending a telegram to Calton to call on him in the morning, stayed indoors all day, and amused himself with smoking and.
	www.termsdefined.net /ps/pseudoprime.html (628 words)

	lucas pseudoprime - OneLook Dictionary Search (Site not responding. Last check: 2007-10-09)
		Tip: Click on the first link on a line below to go directly to a page where "lucas pseudoprime" is defined.
		Lucas Pseudoprime : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include lucas pseudoprime: euler lucas pseudoprime, somer lucas pseudoprime, strong lucas pseudoprime
	www.onelook.com /?w=lucas+pseudoprime (86 words)

	WAS Frobenius, so good? (Site not responding. Last check: 2007-10-09)
		the Lucas sequence is formed using p and q.
		Assuming b=1 for FR (the fastest case), both FR and BW make use of the same algorithm to compute the Lucas sequence.
		Lucas pseudoprime test with method A*) in C++ code.
	www.groupsrv.com /science/post-605909.html (2192 words)

	Documentation for C# BigInteger Class (Site not responding. Last check: 2007-10-09)
		Determines whether this BigInteger is probably prime using a combination of base 2 strong pseudoprime test and Lucas strong pseudoprime test.
		Returns True if this is both a base 2 strong pseudoprime and a strong Lucas pseudoprime.
		Williams, "Édouard Lucas and Primality Testing", Canadian Mathematical Society Series of Monographs and Advance Texts, vol.
	www.trapexit.org /uc/jungerl/lib/otp.net/Otp/Erlang/BigIntegerDoc.html (1552 words)

	Number Theory Glossary
		Rephrased, this says that the order of b in the group of integers modulo p divides (p-1).
		A more general function which is defined for non-prime values of p is the Jacobi symbol.
		Also called the strong pseudoprime test, this test was originally proposed by M. Rabin in Algorithms and Complexity, J.
	www.unf.edu /~mdedeo/Glossary.html (1010 words)

	SOAR Spring 2003 Course (All About Numbers)
		On the other hand, a Lucas pseudoprime is a composite number that is not caught by this Lucas test.
		We checked some numbers by hand and saw that 341 is a base 2 pseudoprime, 121 is a base 3 pseudoprime, 217 is a base 5 pseudoprime, and 373 is, in fact, simply prime.
		A number n that is a base b pseudoprime for every base b relatively prime to n is called a Carmichael number.
	www.math.toronto.edu /mathnet/SOAR2003/Spring (3611 words)

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