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	Primality test - Wikipedia, the free encyclopedia
		A primality test is an algorithm for determining whether an input number is prime.
		Since compositeness is an NP-problem, usual randomized primality tests never report a prime number as composite, but it is possible for a composite number to be reported as prime (for a small fraction of potential witnesses).
		The Miller-Rabin primality test and Solovay-Strassen primality test are more sophisticated variants which detect all composites (once again, this means: for every composite number n, at least 2/3 of numbers a are witnesses of compositeness of n).
	en.wikipedia.org /wiki/Primality_test (941 words)

	Primality test -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-20)
		A primality test is an (A precise rule (or set of rules) specifying how to solve some problem) algorithm for determining whether an input number is (A number that has no factor but itself and 1) prime.
		It is important to note the difference between primality testing and (Click link for more info and facts about integer factorization) integer factorization — factorization is, as of 2005, a computationally hard problem, whereas primality testing, as shown below, is comparatively easy.
		Since compositeness is an (A registered nurse who has received special training and can perform many of the duties of a physician) NP-problem, usual randomized primality tests never report a prime number as composite, but it is possible for a composite number to be reported as prime (for a small fraction of potential witnesses).
	www.absoluteastronomy.com /encyclopedia/p/pr/primality_test.htm (1401 words)

	Directory - Science: Math: Number Theory: Prime Numbers: Primality Tests (Site not responding. Last check: 2007-08-20)
		Probabilisitc tests yield an answer with high probability of correctness, primality proofs yield an answer which is certain to be correct, often with a certificate of primality which can be verified more easily than the original proof.
		Primality Testing · cached · The problem of primality testing and factorization are two distinct problems.
		Primality Testing Applet · cached · A small Java applet to interactively perform strong probable primality tests.
	www.incywincy.com /default?p=793027 (269 words)

	Primality Proving Section Four "The General Purpose Tests" (Site not responding. Last check: 2007-08-20)
		This is one of four chapters on finding primes and proving primality.
		The second chapter discusses finding small primes and the basic probable primality tests.
		The third chapter cover the classical primality tests that have been used to prove primality for 99.99% of the numbers on the largest known prime list.
	primes.utm.edu /prove/prove4.html (84 words)

	Amazon.ca: Books: Primality Testing and Integer Factorization in Public-Key Cryptography (Site not responding. Last check: 2007-08-20)
		Primality testing and integer factorization, as identified by Gauss in his "Disquisitiones Arithmeticae", Article 329, in 1801, are the two most fundamental problems (as well as the two most important research fields) in computational number theory.
		Primality Testing and Integer Factorization in Public-Key Cryptography introduces various algorithms for primality testing and integer factorization, with their applications in public-key cryptography and information security.
		Primality Testing and Integer Factorization in Public-Key Cryptography is designed for a professional audience composed of researchers and practitioners in industry.
	www.amazon.ca /exec/obidos/ASIN/1402076495 (414 words)

	Primality Testing
		In 1976, G. Miller proposed a primality test, which was justified using a generalized form of Riemann's hypothesis.
		The test is justified rigorously, and for the first time ever in this domain, it is necessary to appeal to deep results in the theory of algebraic numbers; it involves calculations with roots of unity and the general reciprocity law for the power residue symbol.
		It was the first primality test in existence that can routinely handle numbers of up 100 decimal digits, and it does so in about 45 seconds.
	db.uwaterloo.ca /~alopez-o/math-faq/node33.html (413 words)

	Lucas's Primality Test With Factored N-1 (Site not responding. Last check: 2007-08-20)
		However, although this congruence is necessary for primality, it isn't quite sufficient, because for any given base b there exist composites N that satisfy (1).
		The basic idea of this test is the foundation for virtually all deterministic primality tests, so it's worthwhile to understand exactly how it works.
		This proves Lucas's primality criterion: If, for some integer b, the quantity b^(N-1) is congruent to 1 modulo N, and if b^((N-1)/q) is NOT congruent to 1 modulo N for ANY prime divisor q of N-1, then N is a prime.
	www.mathpages.com /home/kmath473.htm (570 words)

	Primality testing and factorization (Site not responding. Last check: 2007-08-20)
		It is believed that a general algorithm for quickly factoring integers with as many as 100 digits does not exist.
		The amazing feature here is that we can show that a number a is composite without actually exhibiting two factors b and c, other than 1 and a, such that a=bc.
		The fact that primality testing is relatively easy, while factoring is presumed to be hard, is the basis of the most sophisticated codes used in the U.S.A.'s security and military establishments today.
	www.math.okstate.edu /~wrightd/4713/nt_essay/node16.html (153 words)

	Example: Testing for Primality
		primality test is based on a result from number theory known as Fermat's Little Theorem.
		In these tests, as with the Fermat method, one tests the primality of an integer n by choosing a random integer a
		This starts from an alternate form of Fermat's Little Theorem, which states that if n is a prime number and a is any positive integer less than n, then a raised to the (n-1)st power is congruent to 1 modulo n.
	mitpress.mit.edu /sicp/full-text/sicp/book/node20.html (1835 words)

	Primality Testing algorithms
		In the past 10-15 years fast primality testing algorithms have been a popular subject of research for many Mathematicians, Cryptographers, and Educators.
		Although I was able to show that the two tests were not mutually exclusive, it does seem to be helpful that we could use the Lucas test as a very strong test for primality.
		In summary, Lucas sequences are very useful for fast and reliable primality testing, with a very small number of failed values.
	www.cs.rit.edu /~kls4584/Primality.htm (1477 words)

	[No title]
		Proving the primality of a given integer is a basic task in number theory.
		Primality proving algorithms give a proof of that fact, called a primality certificate.
		In brief, a decreasing sequence of primes is built, the primality of the successor in the list implying that of the predecessor.
	www.lix.polytechnique.fr /~morain/Prgms/ecpp.english.html (525 words)

	Primes and Primality Testing (Site not responding. Last check: 2007-08-20)
		Primality testing algorithms enable the user to certify the primality of prime integers.
		Proving the primality of very big integers can be time consuming and therefore in some of the algorithms using primes and factorization of integers the user can speed up the algorithm by explicitly allowing Magma to use probable primes rather than certified primes.
		If this is the case, the prime p and the exponent k are also returned, Note that the primality of p is rigorously proven.
	www.math.niu.edu /help/math/magmahelp/text521.html (1074 words)

	Factoring and Primality Testing (Site not responding. Last check: 2007-08-20)
		Although factoring and primality testing are related problems, algorithmically they are quite different.
		Implementations: My first choice for factoring or primality testing applications would be PARI, a system capable of handling complex number-theoretic problems on integers with up to 300,000 decimal digits, as well as reals, rationals, complex numbers, polynomials, and matrices.
		The Miller-Rabin [Mil76, Rab80] randomized primality testing algorithm eliminates problems with Carmichael numbers, which are composite integers that always satisfy Fermat's theorem.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE143.HTM (1004 words)

	Wolfram Research, Inc.
		A certificate of primality is a relatively short set of data that can be easily used to prove primality.
		The word "easily" means that using the data to prove primality is much easier and faster than generating the data in the first place.
		This primality proving algorithm is suboptimal if the number is within the range of efficient factoring algorithms.
	documents.wolfram.com /v3/AddOns/NumT_PrimeQ-.html (785 words)

	11Y05: Factorization and primality testing (Site not responding. Last check: 2007-08-20)
		Proth's primality theorem and candidate primes generalizing the Fermat primes
		Note that this is in principle (and -- so far -- in practice) easier than determining the factors of a number shown to be composite.
		A short discussion of the state of the art of primality testing.
	www.math.niu.edu /~rusin/known-math/index/11Y05.html (611 words)

	Primality proving (Site not responding. Last check: 2007-08-20)
		My Ph.D. thesis (co-written with Marc-Paul van der Hulst, under supervision of H.W. Lenstra, Jr., P. van Emde Boas, and A.K. Lenstra) concerns improvements to the general primality proving method using Jacobi sums, to which the names of Adleman, Pomerance, Rumely, Cohen and Lenstra are usually attached.
		Explicit starting values were found (using Magma) for all cases with h less than 100000, with the exception of those h that are of the form 2^m - 1, for which it is proved that a finite solution in the above sense does not exist.
		In the `scriptie' for my `kandidaatsexamen' I considered certain analogues of the primality tests of Lucas-Lehmer type as described above using elliptic curves with complex multiplication.
	www.math.ru.nl /~bosma/research/nuth/prit.html (265 words)

	Detecting False Reports In Primality Tests By The Oddcomp(z) Method
		If a primality test reports a probable prime whilst the Oddcomp(z) method reports a composite, then this is classified as a false report.
		The method uses a conventional primality test to compute probable primes in a finite contiguous Odd(z) sequence and tests the terms against a similar stretch generated by Oddcomp(z).
		Therefore if a primality test reports an integer as a prime and the Oddcomp(z) method reports it as a composite, then the latter is correct thus revealing a false report by the former.
	web.singnet.com.sg /~huens/paper23.htm (1838 words)

	Science - Math - Number Theory - Prime Numbers - Primality Tests - Newsletter - News - Reviews - Education - Ratings
		A Lucas sequence is a sequence of integers characterized by two parameters, P and Q. In practice Q is always 1 and the sequence is taken modulo a large integer....
		Detecting False Reports In Primality Tests By The Oddcomp(z) Method There are more primes than the number of atoms in the universe [1,2 All standing trees on planet Earth will not be sufficient to produce paper pulp to publish all the 512-bit primes in bound volumes.
		Contrast primality tests Contrast various tests for primality Minjae Kim Department of Computer Science Illinois Institute of Technology December 4, 2002 Abstract I present there are various tests for primality developed for a long time.
	www.newsletter-library.com /Science/Math/Number_Theory/Prime_Numbers/Primality_Tests (632 words)

	Lucas Sequences and Primality testing in Cryptography (Site not responding. Last check: 2007-08-20)
		For this paper I would like to implement the algorithm that tests for primality using lucas sequences and compare to our other probablistic primality tests.
		It seems possible that different tests for primality may not have many overlaping "wrong answers" as we seemed to have for the Solovay-Strassen and Miller-Rabin tests.
		I would also like to test other primality testing algorithms, both deterministic (non - polynomial) and probablistic and compare relative advantages and disadvantages.
	www.cs.rit.edu /~kls4584/abstract.html (218 words)

	Phaos Crypto 3.0: Class Prime
		This class implements methods for randomized primality testing and generation.
		as the threshold for primality testing (that is, with a probability of error is less than 1/2^c.)
		bits (that is, most significant bit equal to 1), using a somewhat conservative threshold of 100 for primality testing (that is, with a probability of error is less than 1/2^100.)
	www.phaos.com /resources/docs/Phaos_Crypto_3.0/apidoc/com/phaos/math/Prime.html (597 words)

	BTech Project Abstract - Towards a deterministic polynomial-time Primality Test by Neeraj Kayal and Nitin Saxena (Site not responding. Last check: 2007-08-20)
		We examine a primality testing algorithm presented in Primality and Identity Testing via Chinese Remaindering: FOCS 1999 and the related conjecture in Prashant and Rajat: BTP-report 2001.
		We show that this test is stronger than some of the most popular tests: the Fermat test, the Solovay Strassen test and a strong form of the Fibonacci test.
		Thus, it is arguably the simplest and yet the strongest test for primality.
	www.cse.iitk.ac.in /research/btp2002/primality.html (157 words)

	Primality Proving.
		The third page covers the classical primality tests that have been used to prove primality for 99.9% of the numbers on the largest known prime list.
		Here is the way we usually use the above results to make a quick primality test: start by dividing by the first few primes (say those below 257); then perform strong primality tests base 2, 3,...
		The generalized Riemann hypothesis is far too complicated for us to explain here--but should it be proven, then we would have a very simple primality test.
	uucode.com /obf/dalbec/alg.html (1387 words)

	The Math
		To verify that a first-time Lucas-Lehmer primality test was performed without error, GIMPS runs the primality test a second time.
		If they do not match, then the primality test is run again until a match finally occurs.
		If there were a bug in the FFT code, then the shifting of the S values ensures that the FFTs in the first primality test are dealing with completely different data than the FFTs in the second primality test.
	www.mersenne.org /math.htm (1506 words)

	Science News: Primality tests: an infinity of exceptions. (id... @ HighBeam Research
		As the consummate escape artist of his generation, magician Harry Houdini was famous for slipping out of what looked like inescapable predicaments -- even when tightly bound, handcuffed, and locked in a trunk.
		The simplest way of determining primality is by trial division -- dividing the given number by every number between 2 and the square root of the given number.
		But because certain numbers slip through, it can't serve as a definitive test of primality.
	www.highbeam.com /library/doc0.asp?DOCID=1G1:12677773&refid=ip_almanac_hf (780 words)

	[No title] (Site not responding. Last check: 2007-08-20)
		Part I: Primality and Complexity (1) Primes provide a source of pseudorandomness for many algorithmic computations.
		(2) The Solavay-Strassen Test for primality is a polynomial-time Monte Carlo algorithm and the Miller-Rabin Test for primality is a polynomial-time Las Vegas algorithm.
		Primality testing is known to be in ZPP the intersection of RP an coRP. SQ 2: An Atlantic City algorithm for primality testing is one which runs in polynomial-time and is correct 3/4 of the time.
	www-cs.engr.ccny.cuny.edu /~csmma/cs5726/probset (599 words)

	Omniseek: /Science & Tech /Math /Number Theory /Prime Numbers
		It is possible to find a same result for p=1 [mod 4], with a new corresponding primality test, a generalization to Cunningham chains of second kind and primality tests of a new type ("in cluster").
		This number has 2196 decimal digits and is the new record for general purpose primality proving (as far as we know).
		The elliptic curve primality proving algorithm (ECPP for short) is a modern method of primality proving that does not require auxiliary factorizations.
	www.omniseek.com /srch/{48263} (617 words)

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