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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 (493 words)

	Pseudoprime
		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) (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001567).
	www.ebroadcast.com.au /lookup/encyclopedia/ps/Pseudoprime.html (422 words)

	Perrin, Jean Baptiste - Hutchinson encyclopedia article about Perrin, Jean Baptiste
		Assuming the atomic hypothesis, Perrin demonstrated how the phenomenon of Brownian motion could be used to derive precise values for Avogadro's number.
		Perrin also contributed to the discovery that cathode rays are electrons.
		Perrin was born in Lille and studied in Paris at the Ecole Normale Supérieure.
	encyclopedia.farlex.com /Perrin,+Jean+Baptiste (214 words)

	Pseudoprime info here at en.90-of-100.info (Site not responding. Last check: 2007-10-31)
		The far-reaching of pseudoprimes be obtainable from Fermat's truncated theorem & hence are hailed Fermat pseudoprimes.
		A denominator x a pseudoprime for outright values of a that are coprime to x is hailed a Carmichael number.
		The smallest Fermat pseudoprime for the grovelling 2 is 341.
	en.90-of-100.info /highway-marine/Pseudoprime (555 words)

	Lucas and Perrin Pseudoprimes
		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).
		Perrin's polynomial has discriminant -23 and is monic in both directions.
	www.mathpages.com /home/kmath127.htm (632 words)

	[No title]
		The edge lengths of successive triangles in this spiral satisfy the Perrin recurrence s[n]=s[n-2]+s[n-3] as well as the recurrence s[n]=s[n-1]+s[n-5], as is apparent from the above figure.
		Perrin's sequence also has the interesting property that its terms are cumulative sums of the sequence itself, i.e., we have s[1]=0 and n-5 s[n] = SUM s[k] for n > 1 k=-3 4-n s[n] = SUM s[-k] for n
		This describes the Perrin sequence A(n) Whenever n is a prime number, it divides A(n) exactly....
	www.math.niu.edu /~rusin/known-math/96/perrin (864 words)

	Perrin numbers (Site not responding. Last check: 2007-10-31)
		The column Mathematical Recreations by Ian Stewart in the June issue of Scientific American discusses the Perrin sequence A(n) with:
		The article quotes an experiment by Steven Arno of the Supercomputing Research Center in Bowie, Md., where a lower bound of 15 digits for the size of the smallest Perrin pseudoprime was obtained in 1991.
		Sifting Richard Pinch's table of the 246683 Carmichael numbers up to 10^16, I counted 150 Perrin pseudoprimes.
	www.ai.univie.ac.at /perrin.html (324 words)

	Free Ebooks of Mazon creek (Site not responding. Last check: 2007-10-31)
		A pseudoprime is a probable prime (an integer which shares a property common to all prime number s) which is not actually prime.
		The usual algorithm to generate prime numbers is to generate random odd numbers and primality test them for primality.
		strong pseudoprime s or Euler-Jacobi pseudoprime s, for which there are no analogues of Carmichael numbers.
	mazon.creek.en.rhot.org (1271 words)

	Perrin Pseudoprime (Site not responding. Last check: 2007-10-31)
		Perrin Sequence 0, 2, 3, 2, 5, 5, 7, 10, 12, 17,...
		Several ``unrestricted'' Perrin pseudoprimes are known, the smallest of which are 271441, 904631, 16532714, 24658561,...
		All have S- Signature, and form the sequence Sloane calls ``restricted'' Perrin pseudoprimes: 27664033, 46672291, 102690901,...
	www.math.sdu.edu.cn /mathency/math/p/p273.htm (175 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)

	Pseudoprime (Site not responding. Last check: 2007-10-31)
		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)

	Perrin's Sequence
		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).
		Further discussion of pseudoprimes relative to various polynomials can be found in the notes Pseudoprimes for x^2 - 4x - 9 Lucas and Perrin Pseudoprimes Symmetric Pseudoprimes
	www.mathpages.com /home/kmath345.htm (511 words)

	Pseudoprime (Site not responding. Last check: 2007-10-31)
		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)

	Pseudoprimes: A Survey Of Recent Results (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		Abstract: this paper, we aim at presenting the most recent results achieved in the theory of pseudoprime numbers.
		This includes Lucas-pseudoprimes and the generalization to sequences generated by integer polynomials modulo N, elliptic pseudoprimes.
		3 Pseudoprimes for higher-order linear recurrence sequences (context) - Gurak - 1990
	citeseer.ist.psu.edu /morain92pseudoprimes.html (579 words)

	pseudoprime - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "pseudoprime" is defined.
		Pseudoprime : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include pseudoprime: frobenius pseudoprime, euler jacobi pseudoprime, fibonacci pseudoprime, perrin pseudoprime, somer-lucas pseudoprime, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=pseudoprime (102 words)

	Plastic number - Wikipedia, the free encyclopedia
		The plastic number (also known as the plastic constant or silver number) is the unique real solution of the equation
		It is the limiting ratio of successive terms of the Padovan sequence and the Perrin sequence, and bears the same relationship to these sequences as the golden ratio does to the Fibonacci sequence.
		The plastic number is also a solution of the equation
	en.wikipedia.org /wiki/Plastic_number (119 words)

	List of web sites about Primality Tests: Pseudoprimes
		Carmichael Numbers and Miller-Rabin Test - A worksheet by Professor Alexandre Borovik defines the two concepts and gives exercises to show the latter is not fooled by the former.
		MathPages: Symmetric Pseudoprimes - A pseudoprime type based on symmetric functions.
		Perrin Pseudoprimes - The first Perrin pseudoprimes and a calculator to test for them.
	www.kub.it /dir/800203 (169 words)

	[No title]
		Of course, Jason might refer to the Perrin “series” instead of “sequence.” In the next section, we will see that geometric sequences can also be examined by considering their associated geometric series.
		The most spectacular property of the Perrin sequence is its effectiveness as a test for primality.
		Indeed, the smallest Perrin pseudoprime is n = 277441 = 5212, which is a factor of P277441.
	www.sci.wsu.edu /math/faculty/detemple/fibonacci.doc (2675 words)

	id:A013998 - OEIS Search Results
		%C A013998 "The column Mathematical Recreations by Ian Stewart in the Jun issue of Scientific American discusses the Perrin sequence [A001608] A(n) with: A(0)=3, A(1)=0, A(2)=2, A(n+1)=A(n-1)+A(n-2).
		On Jul 3rd, 1996, I was able to find the two smallest Perrin pseudoprimes:" - Holzbaur %D A013998 W. Adams and D. Shanks, Strong primality tests that are not sufficient, Math.
		%H A013998 Christian Holzbaur, Perrin pseudoprimes a> %H A013998 E. Weisstein, Link to a section of The World of Mathematics. %H A013998 Index entries for sequences related to pseudoprimes %Y A013998 Cf.
	www.research.att.com /~njas/sequences/?q=id:A013998&p=1&n=10&fmt=2 (283 words)

	[No title]
		} { 20 May 1990 } This is described in the article "THE PSEUDOPRIMES TO 25*10^9" by Carl Pomerance, J. Selfridge, and Samuel S. Wagstaff, Jr., Math.
		PROCEDURE Squf(n : number; VAR f : number); { Implements the square forms factorization method of Shanks, using the numerators of the convergents instead of a second continued fraction.
		TestPrime first uses a base-2 strong pseudoprime test, then the Shanks-Adams-Perrin sequence test, and then checks the six composites less than 5*10^10 which pass both tests.
	archives.math.utk.edu /software/msdos/number.theory/malm/malm.readme (2415 words)

	[No title]
		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.
		But if Q turns out to be 1, the pair (5, 5) is used instead.
		But if Q turns out to be 1, the pair (5,5) is used instead.
	www.macalester.edu /~bressoud/books/CNT.m (4172 words)

	www.cpp-home.com : View topic - Fibonacci setup
		I spent a lot of time dealing with sequences defined by recursion, so I am pretty familiar with these things, but I remember it being puzzling at the beginning.
		For the primality test, what my code does is check if a number is simultaneously a Perrin pseudoprime and a Lucas pseudoprime.
		Here is one link that is easy to understand:
	www.cpp-home.com /forum/viewtopic.php?p=63933 (561 words)

	What's New on the Mathematics Archives
		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)

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