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Topic: Extra strong Lucas pseudoprime

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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)

	Baillie-PSW Primality Test
		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).
		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)

	Extra 200 - Encyclopedia Glossary Meaning Explanation Extra 200 (Site not responding. Last check: 2007-10-09)
		Designer Walter Extra introduced it in 1996, and the aircraft is fully certificated in the US to +/-10 g's.
		The 200 is a little smaller than the Extra 300, and is powered by a 200 hp, rather then a 300 hp Lycoming engine.
		However, with the throttle open, it exhibits performance similar to the Pitts S-2– the Pitts has a 60 hp edge, but the aerodynamic and clean design of the Extra makes it comparable in performance.
	www.encyclopedia-glossary.com /en/Extra-200.html (243 words)

	Math Games: Prime Megagap (Site not responding. Last check: 2007-10-09)
		Assuming a finite number of primes leads to a contradiction, due to these extra primes.
		That means they passed strong primality testing (a 2-spsp test, a 3-spsp test, and a Lucas pseudoprime test).
		Unfortunately, it is possible that the strong primality test might fail on some number.
	www.maa.org /editorial/mathgames/mathgames_01_25_04.html (862 words)

	Final Answers - Science - NUMERICANA
		91 is a pseudoprime to half of the bases coprime to itself.
		Strong pseudoprimes to base a are less common than Euler pseudoprimes.
		Counting the bases to which a composite number is a pseudoprime.
	home.att.net /~numericana/answer (5261 words)

	Pseudoprime - Encyclopedia Glossary Meaning Explanation Pseudoprime (Site not responding. Last check: 2007-10-09)
		Here you will find more informations about Pseudoprime.
		base-2 Euler pseudoprimes** (sequence A006970 in OEIS)
		base-2 strong pseudoprimes** (sequence A001262 in OEIS)
	www.encyclopedia-glossary.com /en/Pseudoprime.html (370 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		ABSTRACT: The Lucas sequence is a generalized form of the Fibonacci sequence, that is, a sequence a1,a2,a3,...
		In addition, Lucas sequences will be used to build a p+1 method of factorization (described by Hugh Williams) which is analogous to the Pollard p-1 method.
		We will see that by coupling this strong pseudoprime test with quadratic residues, we arrive at yet an even more powerful method for determining whether a given number is prime.
	www.skidmore.edu /wdb/php.cgi/academics/dvella/showpart.2000.htm (11659 words)

	[No title]
		Then $n$ is a {\it strong pseudoprime to the base $b$} if either (i) $b^{m} \equiv 1 \pmod{n}$ or (ii) $b^{2^{j}m} \equiv -1 \pmod{n}$ for some $j \in [0,s-1]$.
		\vskip.05in\hskip 5pt $\bullet$ \, Strong pseudoprimes base $b$ are pseudoprimes base $b$.
		\vskip.05in\hskip 5pt $\bullet$ \, There are no $n$ which are strong pseudoprimes to every base $b$ with $1 \le b \le n$ and $\gcd(b,n) = 1$.
	www.math.sc.edu /~filaseta/gradcourses/Math788Mnotes (3262 words)

	Reference Database (Site not responding. Last check: 2007-10-09)
		Abstract: The proliferation of probable prime tests in recent years has produced a plethora of definitions with the word "pseudoprime" in them.
		Examples include pseudoprimes, Euler pseudoprimes, strong pseudoprimes, Lucas pseudoprimes, strong Lucas pseudoprimes, extra strong Lucas pseudoprimes and Perrin pseudoprimes.
		One aim of the reformulation is to enable the creation of stronger tests; another is to aid in proving results about large classes of pseudoprimes.
	primes.utm.edu /references/refs.cgi?long=Grantham2000 (168 words)

	sci.crypt: Re: WAS Frobenius, so good?
		>>>> extra strong Lucas pseudoprime test with method A*) in C++ code.
		I say "used" because the very slow BW test can now be safely replaced by the
		In fact, the only reason for which I need the Lucas test is to do MR + WD
	www.derkeiler.com /Newsgroups/sci.crypt/2005-01/0911.html (519 words)

	What's New on the Mathematics Archives (Site not responding. Last check: 2007-10-09)
		The following link was added to the Number Theory section of the Math Archives' Topics in Mathematics:
		Chinese Remainder Theorem, Lucas, Strong Lucas, and Extra Strong Lucas tests, Perrin test, simple pseudoprime test, Solovay-Strassen, and the DSMR test (i.e., strong test, vario usly attributed to R. Dubois.
		Miller, and M.O. Rabin), Atkin's test for n 5 mod 8, Davenport's improved strong test, Cornacchia's algorithm, extended Euclidean algorithm, Fermat factorization, Jacobi symbol computation, fast square detection and integer square roots, square roots mod p (Shanks's RESSOL, Lehmer's method using Lucas sequences, and Peralta's algorithm)
	archives.math.utk.edu /whatsnew/mar99.html (1047 words)

	Of Making Compilations There Is No End
		Do you know what the Rigby points of a triangle are?
		Do you know what the realizer of a poset is? Do you know what an extra-strong Lucas pseudoprime is? Is Laspeyres' index on the tip of your tongue?
		Would you recognize the Golomb-Dickman constant if you saw its first four digits?
	www.siam.org /siamnews/03-00/compile.htm (1267 words)

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