Lucas prime - Factbites

Lucas prime - Factbites
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Topic: Lucas prime

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	L8.html
		Numbers of the form 2^k-1 are called Mersenne numbers, and prime numbers of the form 2^k-1 are called Mersenne primes.
		It looks like the Mersenne exponent k, which creates the Mersenne prime 2^k-1 must also be a prime number.
		A Fermat prime is a Fermat number that is also a prime number.
	www.math.sfu.ca /~gfee/Math342/L81.html (1049 words)

	Will Edgington's Mersenne Page
		Prime exponent Mersenne numbers for which at least one LL test has been run and for which I have some factoring data but no factor are in DB.nf.
		The status of Mersenne numbers with a Mersenne prime exponent (that is, M(M(p)) where M(p) = 2↑p - 1 is a Mersenne prime) is in MMPstats.txt.
		The exponents of known Mersenne primes are in primeM.txt ; the complete factorizations known to me are in factoredM.txt (the largest prime factor is almost always implied, as some of them are _very_ large), and the roughly 6.7 MB lowM.txt contains the known information for all other Mersenne numbers with exponents less than 200,000.
	www.garlic.com /~wedgingt/mersenne.html (3423 words)

	lucas
		At the end of this paper it is conjectured that if n is a strong base 2 pseudoprime and a Lucas probable prime, then n is in fact a prime.
		At the end of this paper * it is conjectured that if n is a strong base 2 pseudoprime and a Lucas * probable prime, then n is in fact a prime.
		At the end of this paper it is conjectured that if n is a * strong base 2 pseudoprime and a Lucas probable prime, then n is in fact * a prime.
	www.numbertheory.org /gnubc/lucas (387 words)

	Mathenomicon.net : Mersenne primes
		was prime, and this was the highest known prime until the mid-twentieth century.
		Several Mersenne primes have been found since then, but not by supercomputers: instead, a remarkable Internet-based project known as G.I.M.P.S. (Great Internet Mersenne Prime Search), which uses spare processor time on thousands of volunteers' personal computers, has found several more Mersenne primes.
		A Mersenne prime is a prime that is of the form
	www.cenius.net /refer/articles/m/mersenneprime_ency/mersenneprime_ency.html (297 words)

	Lucas-Lehmer
		The Lucas-Lehmer test is the mathematical basis for the GIMPS distributed processing project initiated by George Woltman.
		From a purely logical perspective, the equivalence is needed or there might be Mersenne primes missed by the test.
		is prime or not prime based on a deterministic calculation.
	www.jt-actuary.com /lucas-le.htm (1409 words)

	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 (418 words)

	Pseudoprime - Wikipedia, the free encyclopedia
		Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (sequence A001567 in OEIS).
		A pseudoprime is a probable prime (an integer which shares a property common to all prime numbers) which is not actually prime.
		strong pseudoprimes or Euler-Jacobi pseudoprimes, for which there are no analogues of Carmichael numbers.
	en.wikipedia.org /wiki/Pseudoprime (471 words)

	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.
		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.
	en.wikipedia.org /wiki/Pseudoprime (418 words)

	SOAR Spring 2003 Course (All About Numbers)
		On the other hand, a Lucas pseudoprime is a composite number that is not caught by this Lucas test.
		We checked some numbers by hand and saw that 341 is a base 2 pseudoprime, 121 is a base 3 pseudoprime, 217 is a base 5 pseudoprime, and 373 is, in fact, simply prime.
		A number n that is a base b pseudoprime for every base b relatively prime to n is called a Carmichael number.
	www.math.toronto.edu /mathnet/SOAR2003/Spring (3611 words)

	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 (418 words)

	SOAR Spring 2003 Course (All About Numbers)
		On the other hand, a Lucas pseudoprime is a composite number that is not caught by this Lucas test.
		We used the strong primality test to show that 2047 = 1 + 2·1023 is not prime, although it is a strong base 2 pseudoprime.
		We checked some numbers by hand and saw that 341 is a base 2 pseudoprime, 121 is a base 3 pseudoprime, 217 is a base 5 pseudoprime, and 373 is, in fact, simply prime.
	www.math.toronto.edu /mathnet/SOAR2003/Spring (3611 words)

	Luke's Marin Mersenne Page
		The 36th known Mersenne prime is a whopper with 895,932 digits.
		It was not until the mid 20th century that Mersenne became known primarily for his Prime Number Conjecture.
		Not quite a million digits long, it is the 37th known Mersenne prime.
	primes.utm.edu /mersenne/LukeMirror/mersenne.htm (1723 words)

	Wiley::Prime Numbers: The Most Mysterious Figures in Math
		prime number theorem and the prime counting function.
		Fermat and primes of the form x² + y².
		Prime Numbers: The Most Mysterious Figures in Math
	eu.wiley.com /WileyCDA/WileyTitle/productCd-0471462349,descCd-tableOfContents.html (146 words)

	lucas
		At the end of this paper it is conjectured that if n is a strong base 2 pseudoprime and a Lucas probable prime, then n is in fact a prime.
		At the end of this paper it is conjectured that if n is a * strong base 2 pseudoprime and a Lucas probable prime, then n is in fact * a prime.
		At the end of this paper * it is conjectured that if n is a strong base 2 pseudoprime and a Lucas * probable prime, then n is in fact a prime.
	www.numbertheory.org /gnubc/lucas (387 words)

	MathGroup Archive: July 2000 [00038]
		The PrimeQ function combines two strong pseudoprime tests (base 2 and base 3) and the Lucas pseudoprime test, with the results correct up to 10^16 (no known counterexamples and all known primes correctly identified).
		For the 2-strong pseudoprime test, for example, we might infer that if n is odd and 2^(n-1) is congruent to 1 (mod 1), then n is prime.
		Chapter 2 (Prime Numbers) is more basic, but includes a review of strong pseudoprime tests, which are based on Fermat's Little theorem: if p is prime then a^(p-1) congruent to 1 (mod p) if gcd(a,p) =1.
	forums.wolfram.com /mathgroup/archive/2000/Jul/msg00038.html (338 words)

	Carmichael Number
		Carmichael number will pass Fermat test, Lucas test, Miller-Rabin test and any other probabilistic prime test.
		Use other different prime test algorithm for the value which passed the prime test algorithm value.
		NOTE: Normally to avoid Carmichael number, developer use many base values (witnesses), but developer use fewer base values (witnesses) to keep the speed of the running test so it will not hurt the performance.
	www.geocities.com /hmaxf_urlcr/carmichael.htm (338 words)

	BigInteger.cs
		Failed primality test\n"); return false; } } //********************************************************************* // Determines whether this BigInteger is probably prime using a // combination of base 2 strong pseudoprime test and Lucas** strong // pseudoprime test.
		The two // pseudoprimes p and q are fixed, but the two RSA keys are generated // for each round of testing.
		If this BigInteger is a // base 2 strong pseudoprime, proceed on to the next step.
	www.idi.ntnu.no /~simonth/ntnu/ideas/200412/bin/BigInteger.cs (2431 words)

	Large number arithmetic in BASIC
		Williams's double-step, Lucas series P+1 method finds factor p of N if all prime factors but one of p±1 are at most B, and a single larger factor is at most B1.
		Lucas pseudoprime test, quick calculation of Fibonacci giants.
		Lucas-Lehmer test for Mersenne numbers 2^ p - 1, with odd prime p.
	www.home.zonnet.nl /vspickelen/Largefiles/LargeInt.htm (2225 words)

	Lucas-Lehmer primality test
		The Lucas-Lehmer primality test is a method of testing the primality of some number n ; it requires that the prime factors of n -1 be known.
		\n* Edouard Lucas \n* Lucas-Lehmer test \n* Prime number
		Conversely, if n is prime, then there exists a primite root modulo n, and any such primitive root will survive both steps of the algorithm.
	encyclopedia.codeboy.net /wikipedia/l/lu/lucas_lehmer_primality_test.html (2225 words)

	The Top Twenty: Generalized Lucas Number
		Prime generalized Lucas numbers are clearly a particular case of prime primitive parts, occurring when n is also a prime.
		Generalized Lucas numbers were introduced in [Lucas1878] and intensively studied in [Carmichael1913].
		Stewart, "On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers," Proc.
	primes.utm.edu /top20/page.php?id=23 (2225 words)

	Number Theory Glossary
		A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases.
		A prime number is a number which has no factors other than 1 (called non-trivial factors).
		A number which is not composite is called prime.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (2225 words)

	Math Forum - Ask Dr. Math
		It's a little slower then finding out whether M is a strong pseudoprime to every base, which is what you'd really have to do to really show that your number is really a prime, and not just some imposter.
		If you find that M is a strong pseudoprime to the base 2 and M is smaller than 2047, then M is prime.
		If you test it again with the base 3, and it's again a strong pseudoprime, then M is a prime provided it's less than 1373,653.
	mathforum.org /library/drmath/view/51512.html (681 words)

	The Lucas-strong base 2 pseudoprime test
		At the end of this paper it is conjectured that if n is a strong base 2 pseudoprime and a Lucas probable prime, then n is in fact a prime, though this is unlikely to be always the case.
		If 1 is returned, then n is a strong base 2 pseudoprime and a Lucas probable prime.
		(See C. Pomerance, J.L. Selfridge, S. Wagstaff Jr., The Pseudoprimes to 25.10
	www.numbertheory.org /php/lucas.html (84 words)

	BigInteger.cs
		Failed primality test\n"); return false; } } //********************************************************************* // Determines whether this BigInteger is probably prime using a // combination of base 2 strong pseudoprime test and Lucas** strong // pseudoprime test.
		The two // pseudoprimes p and q are fixed, but the two RSA keys are generated // for each round of testing.
		If this BigInteger is a // base 2 strong pseudoprime, proceed on to the next step.
	www.idi.ntnu.no /~simonth/ntnu/ideas/200412/bin/BigInteger.cs (2431 words)

	Finding primes & proving primality
		Rather than divide by just the primes, it is sometimes more practical to divide by 2, 3 and 5; and then by all the numbers congruent to 1, 7, 11, 13, 17, 19, 23, and 29 modulo 30--again stopping when you reach the square root.
		For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens?) The first number is 3 so it is the first odd prime--cross out all of its multiples.
		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.
	www.isk.kth.se /kursinfo/6b2025/litterature/finding_primes.html (2205 words)

	LucasLehmer.frink
		// // To be a Mersenne prime, n must be prime.
		// Returns true if 2^n - 1 is prime, false otherwise.
		This function does not first // test that, as it is assumed that some sort of preliminary sieving has // occurred.
	futureboy.homeip.net /frinksamp/LucasLehmer.frink (138 words)

	Mersenne Primes page - How are they found?
		Mersenne primes can be found using the Lucas-Lehmer test which Lucas began working on in the late 1870's and Lehmer simplified in about 1930:
		The primes q such that 2q + 1 is also a prime are now called Sophie Germain primes as these were studied by Sophie Germain in 1825.
		The following theorem was stated by Euler in 1750 and proved by Lagrange and then by Lucas:
	people.bath.ac.uk /cme20/howfound.html (236 words)

	Mersenne prime
		Mersenne primes rank among the largest of all known primes because they have a particularly simple test for primality, called the Lucas-Lehmer test.
		This changed in 1995 when the American computer scientist George Woltman began the Great Internet Mersenne Prime Search (GIMPS) by providing a database of what exponents had been checked, an efficient program based on the Lucas-Lehmer test that could check these numbers, and a way of reserving exponents to minimize the duplication of effort.
		Around the time that Mersenne's conjecture was finally settled, in 1947, digital computers gave a new impetus to the hunt for Mersenne primes.
	www.daviddarling.info /encyclopedia/M/Mersenne_prime.html (313 words)

	Lucas, (Francois) Edouard (Anatole) (1842-1891)
		He devised methods of testing for prime numbers — work that was later refined by D. Lehmer to yield the Lucas-Lehmer test for checking Mersenne numbers to see if they are prime.
		Lucas was also interested in recreational mathematics, the Tower of Hanoi being his best known puzzle game.
		A French mathematician well known for his study of the Fibonacci sequence and the related Lucas sequences named after him.
	www.daviddarling.info /encyclopedia/L/Lucas.html (160 words)

	gmp prime testing
		Would still be=20 usefull to have a fn to calculate lucas fn's mod n, as these are very co= mmon=20 in prime testing, also the rqft test includes all the underlying code=20 allready.
		> > > I have temporaliry dropped doing the lucas > > probable prime tests for the reason that I guessed they would not get > > accepted and also there is no consistent definition in the books !!!!
		prime= =20 testing, there are slower and less accurate than rqft, but they are fai= rly=20 useful.
	www.swox.com /list-archives/gmp-devel/2002-November/000014.html (718 words)

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