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Topic: Fibonacci pseudoprime

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Euler pseudoprime

Carmichael number

Modular arithmetic theory

Modular arithmetic

Coprime

	pseudoprime (Site not responding. Last check: 2007-10-21)
		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).
	www.yourencyclopedia.net /Pseudoprime.html (538 words)

	Knowledge King - Pseudoprime (Site not responding. Last check: 2007-10-21)
		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 smallest pseudoprime for the base 2 is 341.
		Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (SIDN A001567).
	www.knowledgeking.net /encyclopedia/p/ps/pseudoprime.html (433 words)

	Pseudoprime -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A pseudoprime is a (Click link for more 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 (Click link for more 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 (Click link for more info and facts about Carmichael number) Carmichael number.
	www.absoluteastronomy.com /encyclopedia/p/ps/pseudoprime.htm (631 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		To: Twin Fibonacci Speudprimes, PI (1986) 253 2.
		To: Triangular Fibonacci Pseudoprimes, PI (1986) 253 4.
		To: Pentagonal Fibonacci Pseudoprimes, PI (1986) 254 5.
	www.mscs.dal.ca /Fibonacci/misc.txt (3184 words)

	Symmetric Pseudoprimes (Site not responding. Last check: 2007-10-21)
		A symmetric pseudoprime relative to f is any composite integer c such that f(z^c)=0 (mod c) for every root z of f.
		For example, the smallest symmetric pseudoprime relative to f(x)=x-2 is 341=(11)(31).
		The smallest symmetric pseudoprime relative to this polynomial that is the product of two splitting primes is N = 2258745004684033 = (27439297)(82317889) I suspect that this is the smallest symmetric pseudoprime of any composition relative to the quintic, but I have not made an exhaustive search.
	www.meta-religion.com /Mathematics/Articles/symmetric_pseudoprimes.htm (610 words)

	Fibonacci pseudoprime - Wikipedia, the free encyclopedia
		In number theory, a pseudoprime is a number that passes some test that all primes pass, but is actually composite.
		Anderson, Peter G. Fibonacci Pseudoprimes, their factors, and their entry points.
		Anderson, Peter G. Fibonacci Pseudoprimes under 2,217,967,487 and their factors.
	www.wikipedia.org /wiki/Fibonacci_pseudoprime (163 words)

	Pseudoprime - Encyclopedia, History, Geography and Biography
		The usual algorithm to generate prime numbers is to generate random odd numbers and test them for primality.
		If the user is willing to tolerate a very small chance that the number found is not a prime number but a pseudoprime, then much faster and simpler tests are possible.
		Probabilistic algorithms such as the Fermat primality test, the Solovay-Strassen primality test, and the Miller-Rabin primality test are refinements of this idea.
	www.arikah.net /encyclopedia/Pseudoprime (461 words)

	Lucas and Perrin Pseudoprimes (Site not responding. Last check: 2007-10-21)
		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.
		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.
		This implies that, in general, a quadratic pseudoprime test imposes two congruence conditions (corresponding to the two elementary symmetric functions of the roots).
	www.mathpages.com /home/kmath127.htm (632 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		We are charged by PSW to find and factor or prove the non-existence of a base-2 (Fermat) pseudoprime, congruent to 2 or 3 modulo 5, which is also a Fibonacci pseudoprime.
		In addition, some kind of loose generational structure might appear with the Fibonacci pseudoprimes as there is with the Carmichaels in the form of factor sharing.
		For base-2, all the generated pseudoprimes are congruent to 1 (mod 5); if this can be tweaked to give numbers of the correct residue, it might be of some help.
	www.math.gatech.edu /~lacey/ump/reu/2003/charis_report.txt (1120 words)

	Search Encyclopedia.com
		A sequence is said to be known if a formula can be given for any particular term using the preceding terms or using its position in the sequence.
		In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia BC, it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations.
		Leonardo da Pisa Leonardo da Pisa: see Fibonacci, Leonardo.
	www.encyclopedia.com /searchpool.asp?target=Fibonacci+pseudoprime (299 words)

	2.0 Symmetric Pseudoprimes Relative to Selected Polynomials
		Therefore, the symmetric pseudoprimes relative to f are identical to the ordinary pseudoprimes relative to f.
		Each of the remaining 67 symmetric pseudoprimes is determinate, but eleven of them (listed in parentheses) can be excluded because they exhibit the wrong congruences for their residue classes (mod 5).
		A composite integer N co-prime to 120 is a symmetric pseudoprime relative to the fifth degree polynomial
	www.mathpages.com /home/kmath003/part2/sympart2.htm (3340 words)

	[No title]
		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/known-math/94/primalty.tst (1695 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		is the largest triangular number in the Fibonacci sequence.
		Fibonacci number begins with the digits in n.
		is the first composite number in the Fibonacci sequence with a prime index.
	home.comcast.net /~tdk500/Specialno.html (6437 words)

	Pseudoprime
		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).
		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)

	Andrzej Rotkiewicz
		Pseudoprime numbers and their generalizations, Student Association of the Faculty of Sciences, University of Novi Sad, Novi Sad 1972, pp.
		On the pseudoprimes with respect to the Lucas sequences, Bull.
		On strong Lehmer pseudoprimes in the case of negative discriminant in arithmetic progressions, Acta Arith.
	www.impan.gov.pl /User/rotkiewi (488 words)

	Number Theory Introduction by Brian Conrey (Site not responding. Last check: 2007-10-21)
		Show that 341 is a pseudoprime to the base 2.
		Pseudoprimes to the base a for any a>2 can be defined analogously.
		A number n which is a pseudoprime to every base coprime to n is called an absolute pseudoprime or a Carmichael number.
	www.castilleja.org /faculty/josh_zucker/gunn/bamo/circles/sanjose/NumberTheory.html (1121 words)

	[Haskell-cafe] NumberTheory library
		This function checks if the second argument is a pseudoprime -- for the first argument (or indeed a prime).
		This function checks if the second argument is a prime or a -- strong pseudoprime for the first argument.
		Testing strong pseudoprimality for the first 200 primes.
	www.haskell.org /pipermail/haskell-cafe/2005-May/009922.html (1136 words)

	Wayne McDaniel Publication Page (Site not responding. Last check: 2007-10-21)
		The Irrationality of Certain Series whose Terms are Reciprocals of Lucas Sequence Terms, Fibonacci Quarterly, 32 (1994), 346-351.
		Diophantine Representation of Lucas Sequences, Fibonacci Quarterly, 33 (1995), 59-63.
		On the Factorization of lucas Numbers, Fibonacci Quarterly (Accepted; to be published in 2000).
	www.cs.umsl.edu /~mcdaniel/publication.html (647 words)

	GAP Forum: Re: Re: problems with IsPrimeInt (Site not responding. Last check: 2007-10-21)
		factor less than 1000 and which are strong pseudoprimes for the base 2.
		Fibonacci number is the first to pass all tests except for the last one
		There are known integers that are pseudoprimes w.r.t.
	www-groups.dcs.st-and.ac.uk /~gap/ForumArchive/Schoener.1/Martin.1/Re__Re__.37/1.html (755 words)

	NMBRTHRY Archives -- April 2002 (#4)
		So, for example, just taking a random implementation of the strong pseudoprime test of Artyuhov and Selfridge (often called the Miller-Rabin test), it is certainly possible that a composite number can be declared prime, yet the same composite number, in most other runs, would be found out as composite.
		To pass this test, the number n must be a strong pseudoprime base 2, and to be a Lucas pseudoprime for a canonical Lucas sequence with the least discriminant D (in absolute value) for which (D/n) = -1.
		No one has ever found a composite number which passes this test, and a prize of $620 is offered for an example, with the prime factorization.
	listserv.nodak.edu /cgi-bin/wa.exe?A2=ind0204&L=nmbrthry&F=&S=&P=387 (633 words)

	United and Alternative Left . Catalonia . 1998 . Spain (Site not responding. Last check: 2007-10-21)
		The 20th century initially brought little peace; colonisation of Western Sahara, Spanish Morocco and Equatorial Guinea was attempted as a substitute for the loss of the Americas.
		In number theory, a pseudoprime is a number that passes some test that all Prime_number primes pass, but is actually composite.
		A Fibonacci pseudoprime is a composite number composite integer n that satisfies the following conditions: # P 0 and Q =...
	www.uk.fraquisanto.net /United_and_Alternative_Left (386 words)

	Perrin's Sequence
		For example, most people are familiar with the spiral of equilateral squares whose edge lengths correspond to the Fibonacci numbers, but less well-known is the spiral of equilateral triangles shown below described in The Golden Pentagon.
		For example, the smallest symmetric pseudoprime relative to the Fibonacci quadratic x^2 - x- 1 is 705, whereas the smallest realtive to Perrin's cubic x^3 - x - 1 is 27664033 = (3037)(9109) as found by Shanks and Adams (using an HP-41C calculator!).
		They noted that none of these pseudoprimes had the signature of a prime p such that Perrin's polynomial is irreducible (mod p).
	www.mathpages.com /home/kmath345.htm (511 words)

	Stephen Pierce's Fibonacci Secrets (Site not responding. Last check: 2007-10-21)
		To his credit, Fibonacci is the person who adopted the decimal system and introduced Arabic arithmetic to the Western world.
		Fibonacci is not really all that complex, and I don't mean to alarm you.
		Your Fibonacci Secrets is a superb trading course explaining exactly how to use technical analysis to identify key support and resistance levels.
	FibonacciSecrets.com (5887 words)

	GAP Forum: Re: problems with IsPrimeInt (Site not responding. Last check: 2007-10-21)
		pseudoprimes for the base 2, to be more precise).
		2971th Fibonacci number is the first to pass all tests except for the
		pseudoprime with respect to all the numbers n_1,...
	www-groups.dcs.st-and.ac.uk /gap/ForumArchive/Hoefling.1/Burkhard.1/Re__prob.7/1.html (729 words)

	Number Theory Glossary
		A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases.
		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.
	www.math.umbc.edu /~campbell/NumbThy/Glossary.html (946 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)

	Pseudoprime (Site not responding. Last check: 2007-10-21)
		Composite'' requirement, calling any number which passes the specified tests a pseudoprime even if it is
		Composite numbers which are pseudoprimes to every base; they are sometimes called
		Strong Pseudoprimes spsp to the base 2, as well as
	www.math.sdu.edu.cn /mathency/math/p/p715.htm (88 words)

	[No title]
		Key in 1350 {fibo} 1351 {fibo} to push two large consecutive Fibonacci numbers on the stack, then press {a/b} to display the decimal expansion of their ratio.
		The Miller-Rabin pseudoprime test uses the detection of such peculiar roots of 1 to recognize composite numbers.
		To demonstrate the principle of public-key encryption, we'll encode the number of the beast 666 with a 256-bit key.
	www.home.versatel.nl /poezenbroek/Largefiles/ReadMe.txt (1587 words)

	Encyclopedia: Fibonacci pseudoprime (Site not responding. Last check: 2007-10-21)
		Anderson, Peter G. http://www.cs.rit.edu/~pga/homepage.html) Fibonacci Pseudoprimes, their factors, and their entry points.
		Anderson, Peter G. http://www.cs.rit.edu/~pga/homepage.html) Fibonacci Pseudoprimes under 2,217,967,487 and their factors.
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Fibonacci-pseudoprime (203 words)

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