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Topic: Fermat primality test

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Carmichael number

Prime number

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PGP

	NationMaster - Encyclopedia: Fermat primality test
		The Miller-Rabin primality test is a primality test A primality test is an algorithm for determining whether an input number is prime.
		Like all probabilistic primality tests, there are values of n that will repeatedly produce liars, thus claiming that n is prime when it is actually composite -- these values are known as strong pseudoprimes In mathematics, a strong pseudoprime is a certain kind of natural number.
		The Miller-Rabin test is strictly stronger than the Solovay-Strassen primality test in the sense the set of strong liars of the Miller-Rabin test is a subset of the set of the Solovay-Strassen primality test.
	www.nationmaster.com /encyclopedia/Fermat-primality-test (809 words)

	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 simplest probabilistic primality test is the Fermat primality test.
	en.wikipedia.org /wiki/Primality_test (941 words)

	Primality Proving 2.3: Strong probable-primality and a practical test
		A test based on these results is quite fast, especially when combined with trial division by the first few primes.
		It has been proven ([Monier80] and [Rabin80]) that the strong probable primality test is wrong no more than 1/4th of the time (3 out of 4 numbers which pass it will be prime).
		Jon Grantham's "Frobenius pseudoprimes" can be used to create a test (see [Grantham98]) that takes three times as long as the SPRP test, but is far more than three times as strong (the error rate is less than 1/7710).
	primes.utm.edu /prove/prove2_3.html (998 words)

	primality test (Site not responding. Last check: 2007-10-31)
		The simplest primality "test" is as follows: Given an input number N, we check each integer k > 1 other than N to see whether N is divisible by k.
		The simplest true primality test is as follows: Given an input number N, we check whether it is divisible by any integer between 1 and N exclusive.
		A more convenient primality test is as follows: Given an input number N, we check whether it is divisible by any integer greater than 1 and less than or equal to the square root of N.
	www.yourencyclopedia.net /primality_test.html (829 words)

	Primality test - Open Encyclopedia (Site not responding. Last check: 2007-10-31)
		The simplest primality test is as follows: Given an input number n, we see if each integer k from 2 to n-1 divides n.
		The Miller-Rabin primality test and Solovay-Strassen primality test are more sophisticated variants which detect all composites; they are often the methods of choice.
		The Lucas-Lehmer test relies on the fact that the if the multiplicative order of some number a modulo n is n-1 for a prime n when a is primitive.
	www.open-encyclopedia.com /Primality_test (675 words)

	Fermat's little theorem - Wikipedia, the free encyclopedia
		Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a prime number, then for any integer a,
		Fermat's little theorem is the basis for the Fermat primality test.
		Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have
	en.wikipedia.org /wiki/Fermat%27s_little_theorem (555 words)

	Primality test - Definition, explanation
		It is important to note the difference between primality testing and integer factorization — factorization is, as of 2005, a computationally hard problem, whereas primality testing, as shown below, is comparatively easy.
		The simplest primality test is as follows: Given an input number n, we see if any integer m from 2 to n-1 divides n.
		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).
	www.calsky.com /lexikon/en/txt/p/pr/primality_test.php (1213 words)

	AKS primality test (Site not responding. Last check: 2007-10-31)
		The AKS primality test (also known as Agrawal-Kayal-Saxena primality test and cyclotomic AKS test) is a deterministic primality-proving algorithm discovered and published by three India n scientists named Manindra Agrawal, Neeraj Kayal and Nitin Saxena in August 6, 2002 in a scientific paper titled "PRIMES is in P".
		Lucas's Primality Test With Factored N-1 Kevin Brown explains the mathematics behind the classical "N-1" primality proving algorithm.
		Primality Testing with Fermat's Little Theorem Test numbers for primality and pseudoprimality in Java.
	www.serebella.com /encyclopedia/article-AKS_primality_test.html (470 words)

	Fermat primality test - Wikipedia, the free encyclopedia
		The Fermat primality test is a probabilistic test to determine if a number is composite or probably prime.
		n), where k is the number of times we test a random a, and n is the value we want to test for primality.
		Although Carmichael numbers are rare, there are enough of them that Fermat's primality test is often not used in favor of other primality tests such as Miller-Rabin and Solovay-Strassen.
	en.wikipedia.org /wiki/Fermat_primality_test (352 words)

	Example: Testing for Primality
		Fermat's Little Theorem: If n is a prime number and a is any positive integer less than n, then a raised to the nth power is congruent to a modulo n.
		The Fermat test is performed by choosing at random a number a between 1 and n-1 inclusive and checking whether the remainder modulo n of the nth power of a is equal to a.
		The Fermat test differs in character from most familiar algorithms, in which one computes an answer that is guaranteed to be correct.
	www-mitpress.mit.edu /sicp/chapter1/node17.html (1830 words)

	The Mathematical Tourist
		Primality testing turns out to be incredibly easy if the hunter doesn't mind making a mistake once in a long while, a result that is good enough for many practical applications but not for mathematical proof.
		Primality testing is also easy when the numbers involved have a special form.
		Fermat, and later Euler, discovered methods that simplified the task of testing for primality in the case of Mersenne numbers.
	members.fortunecity.com /templarser/tourist2c.html (2255 words)

	Primality test (Site not responding. Last check: 2007-10-31)
		The simplest primality "test" is as follows: an input number N we check each integer k > 1 other than N to see whether N is divisible by k.
		A more convenient primality test is as Given an input number N we check whether it is divisible any integer greater than 1 and less or equal to the square root of N.
		The Lucas-Lehmer test relies on the fact the if the multiplicative order of some number a modulo n is n -1 for a prime n when a is primitive.
	www.freeglossary.com /Primality_test (1081 words)

	Lucas's Primality Test With Factored N-1 (Site not responding. Last check: 2007-10-31)
		Fermat's Little Theorem assures us that if N is a prime then b^(N-1) = 1 (mod N) (1) for every integer b coprime to N. In contrast, if N is composite it is quite rare for the above congruence to be satisfied for ANY b.
		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.
	mathpages.com /home/kmath473.htm (570 words)

	Fermat number Summary
		A test devised by Pepin in 1877 allows the primality of a Fermat number to be tested relatively quickly on a computer, even though these numbers get huge very quickly.
		Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime.
		Pépin's test is necessary and sufficient test for primality of Fermat numbers which can be implemented by modern computers.
	www.bookrags.com /Fermat_number (1320 words)

	Miller-Rabin primality test - Wikipedia, the free encyclopedia
		The Miller-Rabin primality test is a primality test: an algorithm which determines whether a given number is prime, similar to the Fermat primality test and the Solovay-Strassen primality test.
		Its original version, due to G. Miller, is deterministic, but it relies on the unproven generalized Riemann hypothesis; M.
		Like all probabilistic primality tests, there are values of n that will repeatedly produce liars, thus claiming that n is prime when it is actually composite -- these values are known as strong pseudoprimes.
	en.wikipedia.org /wiki/Miller-Rabin_primality_test (751 words)

	Lucas's Primality Test With Factored N-1 (Site not responding. Last check: 2007-10-31)
		Fermat's Little Theorem assures us that if N is a prime then b^(N-1) = 1 (mod N) (1) for every integer b coprime to N. In contrast, if N is composite it is quite rare for the above congruence to be satisfied for ANY b.
		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.
	www.mathpages.com /home/kmath473.htm (570 words)

	Example: Testing for Primality
		Fermat's Little Theorem: If n is a prime number and a is any positive integer less than n, then a raised to the nth power is congruent to a modulo n.
		The Fermat test is performed by choosing at random a number a between 1 and n-1 inclusive and checking whether the remainder modulo n of the nth power of a is equal to a.
		The Fermat test differs in character from most familiar algorithms, in which one computes an answer that is guaranteed to be correct.
	mitpress.mit.edu /sicp/chapter1/node17.html (1830 words)

	Number theory algorithms
		(A test n^2:=Mod(1,24) guarantees that n is not divisible by 2 or 3.
		They found that the strong primality test sometimes (rarely) passes on composite numbers n for more than 1/8 of all bases xCarmichael number.
		Fermat numbers 2^2^k-1 or generally numbers of the form r^s+a where s is large but r and a are very small integers.
	homepage.mac.com /yacas/manual/Algochapter2.html (3947 words)

	C# BigInteger Class - The Code Project - C# Programming
		This is known as probabilistic primality testing and numbers that passes the test are known as pseudoprimes.
		The test was repeated with 100 different values of a, e, n, and the average time required for each exponentiation was calculated.
		This test works based on the assumption that it is extremely rare for a composite number to be both a base 2 strong pseudoprime and a strong Lucas pseudoprime.
	www.codeproject.com /csharp/biginteger.asp (2778 words)

	Miller-Rabin primality test (Site not responding. Last check: 2007-10-31)
		The Miller-Rabin test is a primality test: an algorithm which determines whether a given number is prime.
		Its original version is probabilistic and similar to the Fermat primality test.
		Suppose n > 1 is an odd integer which we want to test for primality.
	www.sciencedaily.com /encyclopedia/miller_rabin_primality_test (274 words)

	Gregory West
		Composites that pass the test of Fermat’s Little Theorem are uncommon, but the fact that Fermat’s theorem did not inspire algorithms that achieved both speed and reliability meant that, to mathematicians, the algorithms were fundamentally flawed.
		Like Fermat’s theorem, a straightforward application of these criteria to determine if a number is prime leads to many subtests of that number, any of which might actually prove that the number is composite.
		This test must either be inconclusive or else sacrifice a practical running time, but it provides flexibility to balance between the time the test would take and the likelihood that the test could give an inaccurate result.
	www.its.caltech.edu /~sciwrite/journal03/west.html (3312 words)

	Fermat's Little Theorem -- from MathWorld
		It is sometimes called Fermat's primality test and is a necessary but not sufficient test for primality.
		A number satisfying Fermat's little theorem for some nontrivial base and which is not known to be composite is called a probable prime.
		Composite numbers known as Fermat pseudoprimes (or sometimes simply "pseudoprimes") have zero residue for some as and so are not identified as composite.
	www.m-brella.be /math/topics/FermatsLittleTheorem.html (560 words)

	Professional Opportunities
		Fast algorithms for primality testing have been of widespread interest to computer scientists since the early 1970s because of the important role they play in many modern cryptosystems.
		In the context of complexity theory, where problems are categorized by their computational hardness, primality testing was not known to be in P, the class of problems solvable by polynomial-time algorithms.
		Factoring is widely believed to be harder than primality testing; still, the only proof of its hardness is that no one has yet found a method.
	www.siam.org /siamnews/09-02/primality.htm (1463 words)

	[No title]
		Fermat surely knew that when n = 1 the two numbers (34 and 20) are not amicable but he failed to mention such in his letter to Mersenne, perhaps dismissing the counterexample as obvious.
		Fermat then announced the solution to the problem: “Clearly, the number of different pairs of squares of which N represents the difference depends on the number of different pairs of odd factors of which N is composed.” [7, p.
		Fermat’s factoring algorithm was the first to improve upon the Sieve of Eratosthanes and it remained the algorithm of choice for hundreds of years to follow.
	www.math.utexas.edu /~narula/fermat.doc (5434 words)

	Solovay-Strassen primality test (Site not responding. Last check: 2007-10-31)
		The Solovay-Strassen primality test is a probabilistic test to determine if a number is composite or probably prime.
		Much like with the Fermat primality test, however, there are liars.
		Therefore, there are no such values that are guaranteed to be liars all the time, like Carmichael numbers are for Fermat's test.
	www.toshare.info /en/Solovay-Strassen.htm (320 words)

	Contrast primality tests (Site not responding. Last check: 2007-10-31)
		To test n for primality (to see if it is prime) just divide by all of the primes less than the square root of n.
		This is the basis of all modern primality tests whether they are as simple as the test above or something as elaborate such as the methods using elliptic curves or number fields.
		By contrasting many tests for primality, in my opinion there is the specific case each theory is applied efficiently.
	www.csam.iit.edu /~cs549/cs549/project/Contrastprimalitytests.htm (5193 words)

	sci.math FAQ: Prime Numbers
		Primality Testing The problem of primality testing and factorization are two distinct problems.
		Deterministic tests versus Probabilistic or Monte Carlo tests Miller's Test 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.
	www.faqs.org /faqs/sci-math-faq/primes (1277 words)

	BTech Project Abstract - Towards a deterministic polynomial-time Primality Test by Neeraj Kayal and Nitin Saxena (Site not responding. Last check: 2007-10-31)
		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)

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