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Topic: Pseudoprimes

Related Topics

Fermat number

Carmichael number

Sieve of Eratosthenes

Euler pseudoprime

Fibonacci pseudoprime

Prime number

Modular arithmetic theory

Coprime

Pseudorandom number generator

Prime Number Shitting Bear

Modular arithmetic

Complexity classes P and NP

Hash table

Probability

Pierre de Fermat

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	Symmetric Pseudoprimes
		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
		Symmetric pseudoprimes tend to be more rare relative to polynomials with larger Galois groups.
	mathpages.com /home/kmath003/kmath003.htm (379 words)

	NationMaster - Encyclopedia: Lucas pseudoprime
		In the specific case of the Fibonacci sequence, where D = 5, the first pseudoprimes are 323 and 377; (5/323) and (5/377) are both −1, the 324th Fibonacci number is a multiple of 323, and the 378th is a multiple of 377.
		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.
	www.nationmaster.com /encyclopedia/Lucas-pseudoprime (234 words)

	Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana
		The most studied pseudoprimes are pseudoprimes to base 2, which have been variously called Poulet numbers, Fermatians, or Sarrus numbers...
		Conversely, a weak pseudoprime that's coprime with the base is a pseudoprime in the usual sense, otherwise this may or may not be the case.
		We may observe that 91 is thus coprime to twice as many bases as it's a pseudoprime to (72 is the Euler totient of 91).
	home.att.net /~numericana/answer/pseudo.htm (2991 words)

	GNU GMP mpz_probab_prime_p Pseudoprimes
		It is characterized by the fact that the performance of k repetitions of Miller's test by mpz_probab_prime_p is insufficient to resolve N as a composite, whereas k+1 repetitions is sufficient.
		As first noted by Clifford Stern (27 July 2007), the specific mpz_spsp pseudoprimes depend to some extent on the specific version of GMP from which mpz_probab_prime_p is compiled.
		However, the global frequency of pseudoprimes is ultimately a consequence of the number of bases k, so the global frequency has not changed significantly.
	www.trnicely.net /misc/mpzspsp.html (1637 words)

	Euler pseudoprime
		Every Euler pseudoprime is also a Fermat pseudoprime.
		It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.
		The absolute Euler pseudoprimes are a subset of the absolute Fermat pseudoprimes, or Carmichael numbers, and the smallest absolute Euler pseudoprime is 561 = 3·11·17.
	www.ebroadcast.com.au /lookup/encyclopedia/eu/Euler_pseudoprime.html (239 words)

	Strong pseudoprime
		In mathematics, a strong pseudoprime is a certain kind of natural number.
		A strong pseudoprime to base a is always an Euler pseudoprime to base a (Pomerance, Selfridge, Wagstaff 1980), but not all Euler pseudoprimes are strong pseudoprimes.
		As Monier and Rabin showed in 1980, a composite number n is a strong pseudoprime to at most one quarter of all bases Carmichael numbers", numbers that are strong pseudoprimes to all bases.
	www.xasa.com /wiki/en/wikipedia/s/st/strong_pseudoprime.html (210 words)

	Euler-Jacobi pseudoprime
		Every Euler-Jacobi pseudoprime is also a Fermat pseudoprime and an Euler pseudoprime.
		There are no numbers which are Euler-Jacobi pseudoprimes to all bases as Carmichael numbers are.
		The table below gives all Euler-Jacobi pseudoprimes less than 10000 for some prime bases a, this table is in the process of being checked and should be used with caution until this notice is removed.
	www.ebroadcast.com.au /lookup/encyclopedia/eu/Euler-Jacobi_pseudoprime.html (387 words)

	math lessons - Pseudoprime
		A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime.
		Pseudoprimes can be classified according to which property they satisfy.
		A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number.
	www.mathdaily.com /lessons/Pseudoprime (414 words)

	Pseudoprimes For x^2 - 4x - 9
		Every one of these pseudoprimes is congruent to a square (mod 13).
		Robert Harley extended the search up to 861,657,343, which is the 181st complete pseudoprime, and still found no pseudoprime with a non-square residue modulo 13.
		It would be interesting to know the smallest complete pseudoprime relative to x^2 - 4x - 9 that is NOT congruent to a square modulo 13.
	mathpages.com /home/kmath183.htm (160 words)

	Answers to questions about factoring and primality testing
		Pseudoprimes and the Miller-Rabin test for primality (day 2) are covered in section 5.2 (192-200).
		Some numbers as pseudoprimes to several bases, some numbers are pseudoprimes to all bases (these are the Carmichael numbers) and some are of course not pseudoprimes at all.
		If we know that for some base, due to its low density, there are only, say, 5 pseudoprimes to that base among the integers that we can represent then that base might be very useful.
	www.rose-hulman.edu /class/ma/holden/Home/Class/Duke/Math128S/notecardans/Carl-answers/mth128s_quest (1505 words)

	About "Symmetric Pseudoprimes (MathPages)" (Site not responding. Last check: )
		Pseudoprimes Based On The Symmetric Functions Of The Roots Of A Polynomial.
		Given a monic polynomial f with integer coefficients, a symmetric pseudoprime relative to f is defined as a composite integer N such that every elementary symmetric function of the Nth powers of the roots of f is congruent (mod N) to the same function of the first powers.
		Basic propositions and computational techniques associated with symmetric pseudoprimes are presented in this paper, illustrated by specific examples relative to selected polynomials of degrees 1 through 5.
	www.mathforum.com /library/view/4346.html (102 words)

	Perrin sequence
		No one has ever found a composite n that divides A(n), but nor has anyone been able to prove that such numbers, known as Perrin pseudoprimes, don't exist.
		In 1991 Steven Arno of the Supercomputing Research Center in Bowie, Maryland, proved that Perrin pseudoprimes must have at least 15 digits.
		The conjecture that no Perrin pseudoprimes exist is important, because the remainder on dividing P(n) by n can be calculated very rapidly.
	www.daviddarling.info /encyclopedia/P/Perrin_sequence.html (324 words)

	Andrzej Rotkiewicz
		In my book Pseudoprime Numbers and Their Generalizations I gave all what was known about pseudoprimes up to 1972.
		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.
	www.impan.gov.pl /User/rotkiewi (488 words)

	About "Symmetric Pseudoprimes (MathPages)" (Site not responding. Last check: )
		Pseudoprimes Based On The Symmetric Functions Of The Roots Of A Polynomial.
		Given a monic polynomial f with integer coefficients, a symmetric pseudoprime relative to f is defined as a composite integer N such that every elementary symmetric function of the Nth powers of the roots of f is congruent (mod N) to the same function of the first powers.
		Basic propositions and computational techniques associated with symmetric pseudoprimes are presented in this paper, illustrated by specific examples relative to selected polynomials of degrees 1 through 5.
	mathforum.org /library/view/4346.html (102 words)

	Lucas and Perrin Probablistic Prime Tests
		A pseudoprime is a composite number that fools the test; the test passes the pseudoprime off as prime while actually the number has two or more factors.
		Perrin Pseudoprimes are composite numbers that pass the Perrin Test; the function returns 'true' when the number is actually NOT a prime number.
		Perrin Pseudoprimes are very rare, fewer are found for this test than is found for many other tests.
	www.lrbcg.com /jtCullen/Math4.htm (1312 words)

	The Pseudoprimes up to 10^13 - Pinch (ResearchIndex)
		There are 38975 Fermat pseudoprimes (base 2) up to 10 11, 101629 up to 10 12 and 264239 up to 10 13 : we describe the calculations and give some statistics.
		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.
		1 Introduction A (Fermat) pseudoprime (base 2) is a composite number N with the property that 2 N \Gamma1 j 1 mod N.
	citeseer.ist.psu.edu /pinch95pseudoprimes.html (513 words)

	Primality Proving 2.2: Fermat, probable-primality and pseudoprimes
		Some early articles call all numbers satisfying this test pseudoprimes, but now the term pseudoprime is properly reserved for composite probable-primes.
		There are 1,091,987,405 primes less than 25,000,000,000; but only 21,853 pseudoprimes base two [PSW80], so Henri Cohen joked that 2-PRP's are "industrial grade primes" [Pomerance84, p5].
		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)

	[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)

	Mathematics of Computation
		to be the smallest strong pseudoprime to all the first
		Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.
		F. Arnault, Constructing Carmichael numbers which are strong pseudoprimes to several bases, J. Symbolic Computation 20 (1995), 151-161.
	www.ams.org /mcom/2003-72-244/S0025-5718-03-01545-X/home.html (520 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)

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