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Topic: Wagstaff prime

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	Prime Numbers -- from Eric Weisstein's Encyclopedia of Scientific Books
		Brillhart, John; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. Jr.; and Tuckerman, B. Factorizations of bn plus or minus 1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed.
		Narkiewicz, W. The Development of Prime Number Theory: From Euclid to Hardy and Littlewood.
		Prime Numbers and Computer Methods for Factorization, 2nd ed.
	www.ericweisstein.com /encyclopedias/books/PrimeNumbers.html (145 words)

	Encyclopedia: Prime number
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
		A probable prime is an integer which, by virtue of having passed a certain test, is considered to be probably prime.
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.
	www.nationmaster.com /encyclopedia/Prime-number (7528 words)

	Wagstaff prime - Wikipedia, the free encyclopedia
		for a prime number p is called a Wagstaff prime; they are related to the New Mersenne conjecture.
		Wagstaff primes are named after mathematician Samuel S. Wagstaff Jr.
		The first few Wagstaff primes (sequence A000978 in OEIS) are:
	en.wikipedia.org /wiki/Wagstaff_prime (104 words)

	Prime number Summary
		For a long time, prime numbers were thought as having no possible application outside of number theory; this changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem or the Diffie-Hellman key-exchange algorithm.
		The prime number theorem says that the proportion of primes less than x is asymptotic to 1/ln x (in other words, as x gets very large, the likelihood that a number less than x is prime is inversely proportional to the number of digits in x).
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as by the p-adic valuations on Q, for every prime number p.
	www.bookrags.com /Prime_number (7095 words)

	Carl Pomerance
		On the largest prime factors of n and n+1, P.
		Pomerance and S.S. Wagstaff, Jr., Congressus Numerantium 37 (1983), 99-117.
		The probability that a random probable prime is composite, S.H. Kim and C. Pomerance, Math.
	www.math.dartmouth.edu /~carlp (2291 words)

	Number crunchers zero in on record-large number
		Wagstaff says the co-authors are scrambling to find a new format for the third edition of their book so that the factors can fit onto a single line of text.
		In addition, Wagstaff says codes based on large factorizations will someday allow consumers to shop with an "electronic wallet," which will consist of a computer disk or a smart card that contains nothing but series of very large numbers coded to ensure that the electronic cash is spent only once.
		Wagstaff and his group are now working to close in on some of the other "most wanted" numbers, which range from 118 digits to 195 digits.
	www.purdue.edu /UNS/html4ever/1997/9705.Wagstaff.number.html (1407 words)

	AngrySparrow » What’s Special About This Number?
		155 is the sum of the primes between its smallest and largest prime factor.
		641 is the smallest prime factor of 225+1.
		2701 is the smallest number n which divides the average of the nth prime and the primes surrounding it.
	www.angrysparrow.com /2007/02/21/whats-special-about-this-number (22621 words)

	Mersenne prime Summary
		In mathematics, a Mersenne prime is a prime number that is one less than a prime power of two.
		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.
	www.bookrags.com /Mersenne_prime (1089 words)

	[No title]
		Wagstaff says the list is comprised of "hard" numbers, those with no known factors that scientists already have spent thousands of hours trying to solve.
		Once the factors to a number are identified, the results are sent back to Wagstaff, who crosses the number off his list and adds a new one.
		O notation measures how fast an algorithm is; it gives an upper bound on the number of operations (to order of magnitude) in terms of n, the number to be factored, and p, a prime factor of n.
	www.lycos.com /info/factoring--number.html (452 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		Kummer's test allows one to solve for each given prime number the problem of whether it is regular or not: For an odd prime number to be regular it is necessary and sufficient that none of the numerators of the first
		Kummer conjectured that there are on the average twice as many regular prime numbers as irregular ones.
		There are 439 regular and 285 irregular prime numbers among the odd numbers smaller than 5500, cf.
	eom.springer.de /I/i052670.htm (418 words)

	Re: Mersenne: this 3/2 conjecture and a result of Wagstaff
		Wagstaff's equation is a vast overestimate for small n, but maybe >better for large n, or an upper bound.
		As a number theorist, I would guess that the reason why e^{gamman) is not such a good approximation (is this really true?) for the Mersenne primes* that we have seen so far is because of lower order terms.
		It is clearly not the case that the exponent of the n-th Mersenne prime is not (3/2)^P{n} or e^(gamma*n), but something like c^{n+o(n)), where "o(n)" is the usual "little-o of n" (lim_{n \rightarrow \infty} o(n)/n = zero (a severe abuse of notation in that limit!).
	www.mail-archive.com /mersenne@base.com/msg02241.html (436 words)

	rimal information,primal (Site not responding. Last check: 2007-11-05)
		In mathematics, a prime number, or primefor short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number.
		The prime ideals of the ring ofintegers are the ideals (0), (2), (3), (5), (7), (11),...
		With this definition, the primes of the field Q of rational numbers are represented by the standard absolute value function (known as the "infinite prime") as well as bythe p-adic valuations on Q, for every primenumber p.
	www.vsearchmedia.com /rimal.html (2216 words)

	New Mersenne conjecture - Wikipedia, the free encyclopedia
		In mathematics, the New Mersenne conjecture (or Bateman, Selfridge and Wagstaff conjecture) is a statement concerning certain prime numbers; it states that for any odd natural number p, if any two of the following conditions hold, then so does the third:
		Therefore it is only necessary to test odd primes to verify the truth of the conjecture.
		Renaud Lifchitz has shown that the NMC is true up to 12,441,900 by systematically testing all odd primes for which it is already known that one of the conditions holds.
	en.wikipedia.org /wiki/New_Mersenne_conjecture (163 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		%e A100276 35-2=13 is prime;* %e A100276 357-2=103 is prime; %e A100276 35711-2=1153 is prime;* %e A100276 3571113-2=15013 is prime.
		A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096170 primes of the form (n^4+1)/2, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.
		%e A098957 a(14)=53 because the 14th prime is 43, or 101011 binary; reverse of 101011 is 110101, or 53 decimal.
	www.research.att.com:9000 /~njas/sequences/eisBTfry00048.txt (4182 words)

	The book of prime number records
		The distribution of prime numbers is closely connected to the location of the zeros of the Riemann zeta function.
		Therefore, it is worthwhile to count the small twin primes and to seek large ones.
		There are two ways that primes may be in arithmetic progressions: They may lie in a given (infinite) arithmetic progression or they may form the entire (finite) progression.
	ega-math.narod.ru /Liv/Wagstaff.htm (1907 words)

	The Top Twenty: Wagstaff
		As part of the Prime Pages and its list of the Largest Known Primes, we keep a list of the 5000 largest known primes (currently those with 92923 digits or more) plus twenty each of certain selected forms.
		Status of the New Mersenne Prime Conjecture Originally by Conrad Curry
		Status of the New Mersenne Prime Conjecture by Renaud Lifchitz
	primes.utm.edu /top20/page.php?id=67 (164 words)

	What's Special About This Number?
		is the largest number with the property that all smaller numbers relatively prime to it are prime.
		is the sum of the primes between its smallest and largest prime factor.
		is the smallest prime which is the sum of the cubes of the first few primes.
	www.stetson.edu /~efriedma/numbers.html (7306 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Sam Wagstaff, the principal maintainer of the tables, is sending a third edition to the publisher this month.
		At that point, several of the remaining wanted numbers were almost sieved, but the linear algebra had been taking 1-2 calendar-weeks per number, pushing their projected completion dates past the August 31 deadline.
		Then Wagstaff announced that Selfridge was on vacation, unable to revise the wanted lists at this time, and gave us approximately one more week to submit results.
	www.crypto-world.com /etc/CunninghamSept2001.txt (1243 words)

	Wagstaff prime - TheBestLinks.com - Mathematics, Mathematician, Prime number, TheBestLinks.com:Find or fix a stub, ... (Site not responding. Last check: 2007-11-05)
		Wagstaff prime - TheBestLinks.com - Mathematics, Mathematician, Prime number, TheBestLinks.com:Find or fix a stub,...
		Wagstaff primes are named after mathematician Samuel S. Wagstaff.
		The first few Wagstaff primes (OEIS A000978 (http://www.research.att.com/projects/OEIS?Anum=A000978)) are:
	www.thebestlinks.com /Wagstaff_prime.html (145 words)

	Prime Numbers and Factoring
		The Great Internet Mersenne Prime Search (GIMPS) A group dedicated to using the resources of the net to find Mersenne primes and to solve other computationally intensive problems in number theory.
		Includes bibliographic references, although they deal mostly with generating probable primes.
		NFS with four large primes: An explosive experiment.
	www.ontko.com /~rayo/primes (561 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		I read briefly that if a prime p has composite Fibonacci number F_p, then F_p is a Fibonacci pseudoprime.
		That is, I looked for primes p with f(p), f(f(p))...all primes, for some length of iteration; it would be interesting to see if we could connect this somehow to Mills' theorem.
		He then reports that a(m)=lcm(a(p_i^e_i)), where the p_i are the prime factors of m.
	www.math.gatech.edu /~lacey/ump/reu/2003/charis_report.txt (1120 words)

	A New Method For The Generation Of Strong Prime Numbers (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		For the generation of keys for this code, two prime numbers are used.
		Amongst the different methods to deal with this problem, we are here interested in generation of certified prime numbers and we present a new method less costly in terms of computation in regard of the other methods of generation for a given size of prime numbers.
		2 the least prime in an arithmetic progression and theorems co..
	citeseer.ist.psu.edu /3815.html (519 words)

	UROP_and_Masters
		For example, suppose we have the set of primes 2, 5, 7, and we want all numbers less than 100 which have only these primes as divisors.
		This project would be to investigate properties of numbers which are smooth with respect to a set of primes, or with respect to multiple sets of primes.
		Let P be the set of primes P = {p : p does not divide M but p - 1 does divide M}.
	www.d.umn.edu /~jgreene/urop/UROP_and_Masters.html (1604 words)

	Wagstaff prime -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a (An integer that has no integral factors but itself and 1) prime number of the form
		for a prime number p is called a Wagstaff prime; they are related to the (additional info and facts about New Mersenne conjecture) New Mersenne conjecture.
		Wagstaff primes are named after (A person skilled in mathematics) mathematician (additional info and facts about Samuel S. Wagstaff Jr.) Samuel S. Wagstaff Jr.
	www.absoluteastronomy.com /encyclopedia/w/wa/wagstaff_prime.htm (94 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Partially Annotated Prime References This is a collection of all the references used by the local prime pages (not an attempt at a complete bibliography).
		The English version is full of mostly harmless typos starting with the first sentence which lists 244497 as prime.) Erdös49 Paul Erdös, "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem," Proc.
		(The twin primes 6797727.215328+/-1 are found on a 486 microcomputer.) FI97 J. Friedlander and H. Iwaniec, "Using a parity-sensitive sieve to count prime values of a polynomial," Proc.
	w3.impa.br /~gugu/mersenne/refs.txt (1938 words)

	Prime Numbers (Site not responding. Last check: 2007-11-05)
		Mersenne primes are primes of the form 2^p - 1.
		Throughout history, the largest known prime has almost always been a Mersenne prime; the period between Brown et al's discovery in August 1989 and Slowinski and Gage's in March 1992 is one of the few exceptions.
		They are also the first known gigantic twin primes (primes with at least 10,000 digits).
	db.uwaterloo.ca /~alopez-o/math-faq/mathtext/node10.html (1159 words)

	sci.math FAQ: Prime Numbers
		From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math Subject: sci.math FAQ: Prime Numbers Date: 17 Feb 2000 22:51:58 GMT Message-ID: <88hu2e$qtn$1@watserv3.uwaterloo.ca> Reply-To: alopez-o@neumann.uwaterloo.ca Summary: Part 3 of 31, New version Archive-name: sci-math-faq/primes Last-modified: February 20, 1998 Version: 7.5 Prime Numbers Largest known Mersenne prime Mersenne primes are primes of the form 2^p - 1.
		For 2^p - 1 to be prime we must have that p is prime.
		Throughout history, the largest known prime has almost always been a Mersenne prime; the period between Brown et al's discovery in August 1989 and Slowinski and Gage's in March 1992 is one of the few exceptions.
	www.faqs.org /faqs/sci-math-faq/primes (1277 words)

	Newsletter - Patty Wagstaff Air Shows : www.PattyWagstaff.com - The Official Patty Wagstaff Website
		Willam A. Anders, the son of a naval officer, was born in Hong Kong in 1933.
		Ridley was among the passengers killed in the 1957 crash of a military transport.
		Wagstaff eagerly went on to earn numerous flight and instructor ratings.
	www.pattywagstaff.com /cgi-bin/pwasnews.pl?display&mode=1&revdis=rdis (3609 words)

	Inside Purdue: Information technology (Site not responding. Last check: 2007-11-05)
		Wagstaff's research in complex mathematical computations was conducted during a two-week experiment during semester break in which computers in instructional labs were used.
		Although other University departments, such as the Engineering Computing Network, have used Condor, the two-week experiment by the OVPIT staff was the first time Condor was used to conduct research computing on instructional lab computers.
		The experiment was remarkably successful, finishing three years' worth of complex mathematical computations for Wagstaff's research in two weeks.
	purdue.edu /periodicals/insidepurdue/ip2002/020129.computer_labs.html (483 words)

	The Mathematical Tourist
		The first number that divides evenly into N is a prime factor.
		The smallest prime factor is a fairly accessible 13,821,503.
		Clearly, trial division is good enough for factoring a large number that happens to be the product of many small primes, but so far, a 200-digit composite that is the product of two 100-digit primes seems immune to any known assault.
	members.fortunecity.com /templarseries/tourist2d.html (1689 words)

	[No title]
		It is NOT assumed that the moduli M[i] are pairwise relatively prime.
		The prime factors and their multiplicities are stored in P[] and M[] respectively.
		UBASIC version: If a number has all prime divisors less than 2^34, and if all but one are less than 2^17, then Tdvd will completely factor it.
	archives.math.utk.edu /software/msdos/number.theory/malm/malm.readme (2415 words)

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