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Topic: Fermat pseudoprime

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	Primality Proving 2.2: Fermat, probable-primality and pseudoprimes
		Fermat's (Little) Theorem: If p is a prime and if a is any integer, then a
		Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes.
		There may be relatively few pseudoprimes, but there are still infinitely many of them for every base a>1, so we need a tougher test.
	primes.utm.edu /prove/prove2_2.html (461 words)

	Pierre Fermat --Great Minds, Great Thinkers
		Fermat worked on number theory while preparing an edition of Diophantus, and the notes and comments thereon contained the numerous theorems of considerable elegance necessary to develop the theory of numbers.
		Fermat is famous for his "Enigma" that was an extension of Pythagorean Theorem, also known as Fermat's Last Theorem.
		Together with René Descartes, Fermat was one of the two leading mathematicians of the first half of the 17th century.
	www.edinformatics.com /great_thinkers/fermat.htm (198 words)

	Symmetric Pseudoprimes
		This is a necessary but not quite sufficient condition for primality, because there are (rare) composites that satisfy this type of test for certain values of c.
		This is why 341 is called a pseudoprime relative to the base 2.
		We could also define symmetric pseudoprimes in terms of Newton's sums for the roots of a polynomial. Let f denote a monic polynomial of degree d with integer coefficients, and let s(k) denote the sum of the kth powers of the roots of f. Lucas observed that if p is a prime then s(p
	www.mathpages.com /home/kmath003/kmath003.htm (379 words)

	Math Forum - Ask Dr. Math
		Continue this until you either find a proof that N is composite, or you decide that you couldn't possibly be unlucky enough to have an N which is a Fermat pseudoprime to all the bases you used.
		These are even rarer than Fermat pseudoprimes with respect to a given base a, but there are known to be infinitely many.
		A number which fails the Strong Compositeness Test with Base a, yet is composite, is called a "strong pseudoprime with respect to base a." Every strong pseudoprime is a Fermat pseudoprime to the same base, but the reverse is false.
	mathforum.org /dr.math/problems/ware12.17.97.html (1346 words)

	C# BigInteger Class - The Code Project - C# Programming
		Other additional features such as modular exponential, modular inverse, pseudoprime generation and probabilistic primality testing are also supported.
		This is known as probabilistic primality testing and numbers that passes the test are known as pseudoprimes.
		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 (2788 words)

	The Prime Glossary: probable prime
		Fermat's little theorem gives us a powerful test for compositeness: Given n (greater than one), choose a positive integer a and calculate a
		There are only 1,092,19,258 base two probable primes less than 25,000,000,000; but only 21,853 of these are pseudoprimes, so the probable prime test base two would fail only 0.0000874% of the time in this region.
		Below are the odd composite probable primes (pseudoprimes) less than 500 for bases 2, 3,..., 20.
	primes.utm.edu /glossary/page.php/PRP.html (310 words)

	Puzzle 239. psp (2) & n^2 -2 numbers
		A composite integer p is said to be a Fermat pseudoprime base a [p is a psp(a)] if it passes a Fermat test for some integer a, that is to say if a
		Therefore, the Fermat test is valid to prove the primality of all the numbers of the form n
		For odd p numbers a preferable congruence pseudoprimality criteria should be: p is a psp(a) if...
	www.primepuzzles.net /puzzles/puzz_239.htm (740 words)

	Fermat.nb
		The converse of Fermat's Little Theorem is false.
		is a pseudoprime to a surprising number of bases.
		In fact, 29341 is a pseudoprime to every prime base other than its prime divisors.
	faculty.uml.edu /klevasseur/math/fermat/Links/Fermat_lnk_2.html (130 words)

	Pseudoprimes and Carmichael numbers
		A Fermat pseudoprime base b is a composite number N which satisfies the Fermat condition b
		The numbers were generated by a variety of strategies, the most important being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique..
		A Carmichael number is a pseudoprime for every possible base b: that is, for every b coprime to N.
	www.chalcedon.demon.co.uk /rgep/carpsp.html (448 words)

	Encyclopédie :: encyclopedia : Test de primalité de Miller-Rabin (Site not responding. Last check: 2007-10-10)
		Le test de primalité de Miller-Rabin est un test de primalité probabiliste : c’est-à-dire un algorithme qui détermine si un nombre donné est probablement premier, de façon similaire au test de primalité de Fermat et le test de primalité de Solovay-Strassen.
		Comme pour les tests de Fermat et de Solovay-Strassen, avec le test de Miller-Rabin nous relions une équation ou un système d'équations qui sont vraies pour des valeurs premières, et nous regardons si elles sont toujours vraies ou non pour un nombre dont nous voulons tester la primalité.
		Pour montrer que l'une d'elles doit être vraie, réutilisons le petit théorème de Fermat :
	www.encyclopedie.cc /Test_de_primalit%c3%a9_de_Miller-Rabin (794 words)

	Math Forum Discussions - Re: Can a strong pseudoprime be a square ?
		Re: Can a strong pseudoprime be a square ?
		Yes, in fact there is a generalization that counts Fermat liars:
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=13&threadID=86906&messageID=424648 (590 words)

	The On-Line Encyclopedia of Integer Sequences
		Smallest base-2 Fermat pseudoprime x that has ord(2,x) = n, or 0 if one does not exist.
		A base-2 Fermat pseudoprime is a composite number x such that 2^x = 2 mod x.
		For a number x to have order n, it must be a factor of 2^n-1 and not be a factor of 2^r-1 for r
	www.research.att.com /~njas/sequences/A086250 (203 words)

	Re: Interesting Discussion with US Government Computer Expert
		Unfortunately, some authors drop the "composite" requirement, calling any number that passes the specified tests a pseudoprime even if it is prime.
		"Pseudoprime" used without qualification means Fermat pseudoprime." I use the term in the meaning of the first sentence.If I refere only to a Fermat pseudoprime, I find it necessary to say Fermat pseudoprime as a specific type of pseudoprime.
		A Miller-Rabin pseudoprime would be a composite that passes a series of Miller-Rabin tests.
	www.newsfeeds.com /archive/sci-crypt/msg01106.html (727 words)

	The Prime Glossary: pseudoprime
		A probable-prime which is composite is called a pseudoprime.
		(At one time all probable primes were called pseudoprimes, but now the terminology has been corrected.) The smallest examples of pseudoprimes for bases 2, 3, 5, and 7 are as follows.
		Jaeschke, "On strong pseudoprimes to several bases," Math.
	primes.utm.edu /glossary/page.php/Pseudoprime.html (321 words)

	Large number arithmetic in BASIC
		Fermat's method is useful only if N has factors close to its square root.
		Pseudoprime test for N using k-th order recurrence relations,
		Lucas pseudoprime test, quick calculation of Fibonacci giants.
	largeint.sourceforge.net (2225 words)

	Fermat Pseudoprime Puzzle
		So m is a base 2 Fermat pseudoprime.
		Since there are infinitely many primes, there are infinitely many base 2 Fermat pseudoprimes.
		The webmaster and author of the Math Help site is Graeme McRae.
	mcraefamily.com /MathHelp/PuzzleFermatPseudoprimeBase2Answer.htm (83 words)

	Search Tuna Report for probable prime code (Site not responding. Last check: 2007-10-10)
		Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography....
		Description: probable prime In number theory, a probable prime (PRP) is an integer that satisfies a condition also satisfied by all prime numbers....
		That means they passed strong primality testing a 2-spsp test, a 3-spsp test, and a Lucas pseudoprime test....
	www.searchtuna.com /ftlive2/3821.html (1149 words)

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