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Topic: Mersenne numbers

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	Mersenne prime - Wikipedia, the free encyclopedia
		Mersenne primes have a close connection to perfect numbers, which are numbers that are equal to the sum of their proper divisors.
		Historically, the study of Mersenne primes was motivated by this connection; in the 4th century BC Euclid demonstrated that if M is a Mersenne prime then M(M+1)/2 is a perfect number.
		The best method presently known for testing the primality of Mersenne numbers is based on the computation of a recurring sequence, as developed originally by Lucas in 1878 and improved by Lehmer in the 1930s, now known as the Lucas-Lehmer test.
	en.wikipedia.org /wiki/Mersenne_prime (784 words)

	Mersenne (Site not responding. Last check: 2007-10-08)
		Mersenne, however, was devoted to study, which he loved, and, showing that he was ready for responsibilities of the world, had decided to further his education in Paris.
		Mersenne thought Galileo's assumption that a falling body passes through infinite degrees of speed was incompatible with a genuinely mechanistic explanation of acceleration.
		Mersenne was doubtful that the air pressure actually supported the mercury and on his return attempted to re-do the experiment but did not have the necessary equipment.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Mersenne.html (2914 words)

	PlanetMath: Mersenne numbers
		Mersenne primes have a strong connection with perfect numbers.
		It is conjectured that the density of Mersenne primes with exponent
		This is version 10 of Mersenne numbers, born on 2001-10-18, modified 2005-03-14.
	planetmath.org /encyclopedia/MersenneNumbers.html (153 words)

	Math Forum: Ask Dr. Math FAQ: Perfect Numbers
		This is because there is a strong link between perfect numbers and a certain kind of prime number (the Mersenne primes).
		Two of these numbers, the 25th and 26th, were found by two high school students in 1978 and 1979.
		Mersenne numbers are named after Marin Mersenne, a French monk.
	mathforum.org /dr.math/faq/faq.perfect.html (2036 words)

	Will Edgington's Mersenne Page
		Counts of Mersenne numbers with at least one known factor with exponent within certain ranges are in 1stfacnt.txt.
		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.
		A "prime-exponent Mersenne number" is a Mersenne number with a prime exponent.
	www.garlic.com /~wedgingt/mersenne.html (3423 words)

	[No title]
		Mersenne primes, as well as Marin Mersenne, have been important to the advancement in mathematics and their properties have brought on many new discoveries in mathematics.
		Mersenne primes have been so widely studied because of the fact that if the exponent is not prime then it is easy to show that the result will be composite (theorem 3.1) and because the largest known prime numbers are usually Mersenne primes.
		Mersenne numbers have the form 2r * s — 1r * s since 1 raised to any power is always one.
	www.saintjoe.edu /~karend/m441/JohnFinalPaper.doc (3216 words)

	Temple University News Bureau - Top Stories (Site not responding. Last check: 2007-10-08)
		His targets are rare, incredibly large numbers known as Mersenne prime numbers, and his weapon is a collection of nearly 300 computers in the Educational Technology Center (ETC) of Temple's College of Liberal Arts.
		Mersenne numbers are named after the French monk and mathematician Marin Mersenne (1588-1648), who stated in the preface to his Cogitata Physico-Mathematica (1644) the 2p - 1 formula, which he believed would yield prime numbers only.
		He explains that among the numbers being tested, the odds of a given number being a Mersenne prime are 1 in 45,000.
	www.temple.edu /news_media/td844.html (911 words)

	Perfect numbers (Site not responding. Last check: 2007-10-08)
		Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
		Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.
		Mersenne was very interested in the results that Fermat sent him on perfect numbers and soon produced a claim of his own which was to fascinate mathematicians for a great many years.
	www-history.mcs.st-and.ac.uk /history/HistTopics/Perfect_numbers.html (4298 words)

	BBC - h2g2 - Mersenne Numbers (Site not responding. Last check: 2007-10-08)
		The study of Mersenne numbers is part of the study of prime numbers.
		Numbers greater than 1 that are not prime are called composite numbers, because they are composed by multiplying primes together.
		A Mersenne number that is prime is known as a Mersenne prime.
	www.bbc.co.uk /dna/collective/A670051 (2025 words)

	Prime numbers (Site not responding. Last check: 2007-10-08)
		The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties.
		A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.
		The statement that the density of primes is 1/log(n) is known as the Prime Number Theorem.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Prime_numbers.html (1589 words)

	Mersenne, Marin -- Encyclopædia Britannica
		While best remembered by mathematicians for his search for a formula to generate prime numbers based on what are now known as “Mersenne numbers,” his wider significance stems from his role as correspondent, publicizing and disseminating the work of some…
		While best remembered by mathematicians for his search for a formula to generate prime numbers based on what are now known as “Mersenne numbers,” his wider significance stems from his role as correspondent, publicizing and disseminating the work of some of the greatest thinkers of his age.
		Most numbers are either “abundant” or “deficient.” In an abundant number, the sum of its proper divisors (i.e., including 1 but excluding the number itself) is greater than the number; in a deficient number, the sum of its proper divisors is less than the number.
	www.britannica.com /eb/article-9052176 (730 words)

	Mersenne Prime Numbers (Site not responding. Last check: 2007-10-08)
		The concept of a Mersenne Prime is evolved from that of a perfect number.
		A perfect number is an integer for which the sum of its divisors is twice the number.
		A Mersenne Number is a number Mm which may or may not be prime.
	www.resort.com /~banshee/Info/mersenne.html (465 words)

	RE: Mersenne: My numbers have been re-assigned to someone else...
		I recall a problem of some sort recently with older versions of the client snagging 20K exponents for factoring even though they aren't capable of factoring ones that big...
		I > have > not released these numbers back, and the program is still working on them > (they are still in the worktodo.ini file).
		Mersenne: My numbers have been re-assigned to someone else...
	www.mail-archive.com /mersenne@base.com/msg07594.html (366 words)

	All Fermat numbers are squarefree (and more)
		Since Fermat numbers are the "least random numbers of all numbers" (a sequence of 0s limited by two 1s), the weight of such arguments has to be considered prudentially.
		Since the powers of 2 modulo any odd number n are units modulo n, in application of Lagrange's Theorem, "The order of a subgroup divides the order of the group", the order of the powers of 2 modulo an odd integer n entirely divides the order of the units modulo n.
		Antisymmetric numbers are numbers whose expression 1/n in binary is a ß/2 long sequence followed by the same sequence inverted.
	www.dybot.com /numbers/sqfree.htm (4373 words)

	getty.net GIMPS Participation
		Mersenne numbers are named after the French monk Marin Mersenne (1588-1648) who stated in the preface to his Cogitata Physica-Mathematica (1644) the definition of what would later be known as a Mersenne Prime Number.
		Each client then works on the number that they were assigned, it determines if it is a prime number or not, then sends the results back to the central server, and requests another number to process.
		Mersenne Primes were found using the PrimeNet central server in conjunction with the GIMPS effort.
	getty.net /gimps (1372 words)

	Numbers
		Two numbers n and m are called an amicable pair if the sum of all positive divisors of n is equal to the sum of all positive divisors of m and both are equal to n + m.
		Carmichael numbers behave like prime numbers with respect to the most useful primality test, that is they pretend to be prime.
		Pentagonal numbers to pentagons is the same as triangular numbers to triangles and square numbers to squares.
	www.tanyakhovanova.com /Numbers/numbers.html (1939 words)

	Mersenne Prime Numbers (Site not responding. Last check: 2007-10-08)
		Mersenne prime numbers are numbers of the form
		Indeed, at present only 33 Mersenne prime numbers are known.
		The largest Mersenne prime number is actually also the largest prime number known.
	www.math.utah.edu /~alfeld/math/mersenne.html.old (114 words)

	The Prime Glossary: Mersenne number (Site not responding. Last check: 2007-10-08)
		Mersenne numbers are integers of the form M
		They are of interest because the Mersenne primes (prime Mersenne numbers) are among the oldest and most studied of all primes!
		These numbers are named after the French monk Mersenne because he encouraged many mathematicians to study them and incorrectly conjectured that the Mersenne numbers were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, and 257; and all the other Mersennes with n < 257 were composite.
	primes.utm.edu /glossary/page.php?sort=MersenneNumber (101 words)

	Biography of Marin Mersenne
		Marin Mersenne is best known for his role as a sort of clearing house for correspondence between eminent philosophers and scientists, and for his work in number theory.
		In 1633, Mersenne published Traite de mouvments, and in 1634 he published Les Mechanique de Galile, which was a version of Galileo's lectures on mechanics.
		It was obvious to Mersenne's peers that he could not have tested all of these numbers (in fact he admitted as much), but they could not test them either.
	www.andrews.edu /~calkins/math/biograph/biomerse.htm (1044 words)

	Large Numbers -- Notes at MROB
		Other sequences, such as the Mersenne numbers are relatively prime, but there is no simple iterative formula to generate them.
		Searches for Mersenne primes are simplified by a number of special tests, that are not generally applicable to searches for all primes.
		Their goal was to make it possible to derive every true theorem in number theory by starting with a set of axioms and a set of inference rules, and methodically applying all the inference rules to the axioms and existing theorems to create new theorems.
	home.earthlink.net /~mrob/pub/math/ln-notes1.html (8135 words)

	Mathematics Archives - Numbers
		In the section on applications there are a number of interactive programs that convert rationals (or quadratic irrationals) into a simple continued fraction, as well as the converse.
		All numbers are not created equal; that certain constants appear at all and then echo throughout mathematics, in seemingly independent ways, is a source of fascination.
		A prime k-tuplet is a sequence of consecutive prime numbers {p1, p2,..., pk} such that, in some sense, pk - p1 is as small as possible.
	archives.math.utk.edu /subjects/numbers.html (1310 words)

	Fermat 6 (Site not responding. Last check: 2007-10-08)
		Here, then, is my discovered linking of the Mersenne and Fermat numbers (if it is already buried out there in the literature, then obviously I apologise for wasting your time in reading this, and I end up with egg on face.
		The Mersenne numbers are the M[p] with M[p] = 2^p - 1, p prime; the Fermat numbers are the F[n] = 2^(2^n) + 1, n = 0, 1, 2, 3, 4, 5,...
		But the Fermat numbers are simply the left vertical side of the above square, and what about the next rank along from it?, and the one alongside it?, and...
	www.spd.dcu.ie /johnbcos/fermat6.htm (1986 words)

	Open Directory - Science: Math: Number Theory: Prime Numbers: Mersenne (Site not responding. Last check: 2007-10-08)
		Marin Mersenne - Biography of the Minim friar and his contributions to number theory.
		Mersenne Prime Digits and Names - Landon Curt Noll lists the decimal, English, and American expansions of all the known Mersenne primes.
		Mersenne Prime Mailing List - Unmoderated list for communication between those interested in all aspects of Mersenne Primes.
	dmoz.org /Science/Math/Number_Theory/Prime_Numbers/Mersenne (237 words)

	Math Trek: Primal Surge, Science News Online, March 5, 2005 (Site not responding. Last check: 2007-10-08)
		At the same time, the discovery of ever-larger Mersenne primes, as these numbers came to be called, steadily progressed to larger and larger values.
		GIMPS volunteers are responsible for checking Mersenne numbers within specified ranges of exponents, whenever their computers would otherwise be idle.
		Posters (with optional magnifier) showing all the decimal digits of the largest known prime numbers can be purchased from Perfectly Scientific at http://www.perfsci.com/novelties.htm.
	www.sciencenews.org /articles/20050305/mathtrek.asp (728 words)

	Mathematics Enrichment Workshop: The Perfect Number Journey (Site not responding. Last check: 2007-10-08)
		A perfect number is then obtained by multiplying this sum to the last power of 2.
		Before you proceed to find the fifth perfect number, you may want to pause for a moment and take a closer look at the first four perfect numbers that have been obtained this way.
		It may be more obvious if we express each number in terms of powers of two, as we shall see in the exercise that follows.
	home1.pacific.net.sg /~novelway/MEW2/lesson1.html (522 words)

	SS > factoids > Mersenne prime
		Mersenne numbers have a particulary simple test for primality, the Lucas-Lehmer test.
		Each Mersenne prime corresponds to an even perfect number.
		Not all the non-prime Mersennes have been completely factored.
	www-users.cs.york.ac.uk /~susan/cyc/m/mersenne.htm (192 words)

	Section 6.2: Mersenne Numbers (Site not responding. Last check: 2007-10-08)
		The first column shows the value of j, the second column indicates whether or not the jth Mersenne number is prime, and the third column shows the value of the jth Mersenne number.
		There is a connection between Mersenne primes and perfect numbers.
		A perfect number is an integer which is equal to the sum of its positive divisors less than itself.
	www.math.uh.edu /~minru/web/special2.html (331 words)

	Glucas - Yet Another FFT
		Glucas is a free program to test primality of Mersenne numbers (numbers with the form 2^n - 1).
		Its results say us whether a Mersenne number is a prime number.
		GIMPS already has found the four biggest primes in the history and is one of the pioneers in the distributed computing in the net.
	www.oxixares.com /glucas (444 words)

	Generalized Mersenne Numbers - Solinas (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		There is a well known shortcut for modular multiplication modulo a Mersenne number, performing modular reduction without integer division.
		We generalize this technique to a larger class of primes, and discuss parameter choices which are particularly well suited for machine implementation.
		Solinas, "Generalized Mersenne numbers", Technical report CORR-39, Dept. of C&O, University of Waterloo, 1999.
	citeseer.ist.psu.edu /solinas99generalized.html (301 words)

	MM61 (Site not responding. Last check: 2007-10-08)
		It is not known whether the double-Mersenne number MM61 = 2^(2^61-1) - 1 is prime or composite, and, just as with other Mersenne numbers, it is interesting to resolve this question one way or another.
		If you have a PC and are interested in helping to find a factor of 2^(2^61-1) - 1: First download the program 'MFAC' and perform a simple test to see that the software works on your computer.
		All other double-Mersenne numbers are 'status unknown', and MM61 happens to be the smallest.
	www.ltkz.demon.co.uk /ar2/mm61.htm (550 words)

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