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Topic: Sophie Germain prime

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Germain primes

Prime number

Cunningham chain

Lucas number

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Kummer extension

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	Encyclopedia article: Sophie Germain prime (Site not responding. Last check: 2007-11-06)
		A prime number (An integer that has no integral factors but itself and 1) p is called a Sophie Germain prime if 2p + 1 is also prime.
		It is conjectured that there are infinitely many Sophie Germain primes, but like the twin prime conjecture (additional info and facts about twin prime conjecture), this has not been proven.
		A sequence of Sophie Germain primes is called a Cunningham chain (additional info and facts about Cunningham chain) of the first kind.
	www.absoluteastronomy.com /encyclopedia/s/so/sophie_germain_prime.htm (251 words)

	Encyclopedia: Prime number
		In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself.
		Prime numbers are of fundamental importance in number theory.
		Prime ideals are an important tool and object of study in commutative algebra and algebraic geometry.
	www.nationmaster.com /encyclopedia/Prime-number (1039 words)

	Sophie Germain - Wikipedia, the free encyclopedia
		Marie-Sophie Germain (April 1, 1776 – June 27, 1831) was a French mathematician, and one of the most important female mathematicians of all time.
		Germain was born to a middle-class merchant family in Paris, France, and began studying mathematics at age thirteen, despite her parents' strong attempts to dissuade her from engaging in a 'men's profession'.
		One significant contribution is the concept of the Sophie Germain prime, which is a prime number p where 2p+1 is also prime.
	en.wikipedia.org /wiki/Sophie_Germain (634 words)

	Sophie Germain prime - Wikipedia, the free encyclopedia
		It is conjectured that there are infinitely many Sophie Germain primes, but like the twin prime conjecture, this has not been proven.
		A sequence {p, 2p + 1, 2(2p + 1) + 1,...} of Sophie Germain primes is called a Cunningham chain of the first kind.
		Every term of such a sequence except the first and last is both a Sophie Germain prime and a safe prime.
	en.wikipedia.org /wiki/Sophie_Germain_prime (284 words)

	Germain (Site not responding. Last check: 2007-11-06)
		Sophie's home was a meeting place for those interested in liberal reforms and she was exposed to political and philosophical discussions during her early years.
		Germain's third attempt in the re-opened contest of 1815 was deemed worthy of the prize of a medal of one kilogram of gold, although deficiencies in its mathematical rigour remained.
		Germain attempted to extend her research, in a paper submitted in 1825 to a commission of the Institut de France, whose members included Poisson, Gaspard de Prony and Laplace.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Germain.html (1029 words)

	Sophie Germain
		Sophie Germain was born in an era of revolution.
		Sophie Germain was born in Paris on April 1, 1776 to Ambroise-Francois and Marie Germain.
		Sophie's interest in mathematics began during the French Revolution when she was 13 years old and confined to her home due to the danger caused by revolts in Paris.
	www.scottlan.edu /lriddle/women/germain.htm (1495 words)

	NOVA Online \| The Proof \| Math's Hidden Woman
		Sophie Germain was born on April 1, 1776 the daughter of a merchant, Ambroise-Francois Germain.
		Germain concluded that if somebody could be so consumed by a geometric problem that it could lead to their death, then mathematics must be the most captivating subject in the world.
		When Germain wrote to Gauss she was still in her 20s, and, although she had gained a reputation in Paris, she feared that the great man would not take her seriously because of her gender.
	www.pbs.org /wgbh/nova/proof/germain.html (2400 words)

	Sophie Germain
		Germain submitted a report on analysis to Lagrange using the name of an acquaintance registered as a student at the school, Antoine-August Le Blanc, or better known as Monsieur Le Blanc, because she felt her answers would not be accepted if it was known that the author was female (…192).
		Germain explained to Gauss of her fear of ridicule because of her sex and the disrepute attached to the femme-savantes of the time.
		Germain, herself, did not have the grasp on double integrals that was necessary for this type of work.
	www.mathsci.appstate.edu /~sjg/womeninmath/SophieGermain.html (3338 words)

	Biograpy of Sophie Germain
		Sophie was not really interested in the study of mathematics until the age of 13 when the French Revolution started.
		Sophie was 18, but she could not enroll in the academy because women were not allowed.
		Sophie Germain was diagnosed with breast cancer and fought over it for most of her life.
	www.andrews.edu /~calkins/math/biograph/199899/biogermn.htm (1272 words)

	Sophie Germain (Site not responding. Last check: 2007-11-06)
		Archimedes: When Sophie was 13, she read about the legend of Archimedes' death: how he was so absorbed in the geometric figure he was drawing in the sand that he ignored the Roman soldier asking him a question and was thus speared to death.
		Sophie was forced to reveal her identity, but Lagrange was impressed by this young woman and became her mentor and friend.
		For example, 5 is a Germain prime because 25+1=11, which is also a prime* number, but 7 is not a prime because 15 (27+1) is not a prime* numbe r.
	www.the4cs.com /~corin/motm/sophie_germain.html (454 words)

	More Prime Patterns
		If P is greater than 2 and is a prime, than if 2P+1 is also prime, P is known as a Sophie Germaine prime.
		All 22 primes in the number range of 1 to 81 are attacked by the queen.
		A prime spiral arranged as a circle is thoroughly discussed and illustrated at Rom Sacks http://www.numberspiral.com/
	www.geocities.com /CapeCanaveral/Launchpad/4057/moreprimes.htm (1277 words)

	Sophie Germain and FLT
		The result now known as Sophie Germain's Theorem was presented in 1823 by Legendre in a paper to the French Academy of Sciences and included in a supplement to his second edition of The Theory of Numbers.
		Legendre generalized Germain's argument to show that properties (1) and (2) hold for the odd prime exponent n provided that one of the numbers 4n+1, 8n+1, 10n+1, 14n+1, or 16n+1 is a prime.
		It is impossible, however, to have a sequence n, p, r of three Sophie Germain primes that are all palindromes because 2r+1 would always end in 5 in such a sequence and thus would not be a prime.
	www.agnesscott.edu /lriddle/women/germain-FLT/SGandFLT.htm (2774 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		For second order Sophie Germain primes, $2(2p+1)+1=4p+3$ is also prime and a similar pattern holds for general $n$-order primes.
		There are 9,592 prime numbers between 1 and 100,000, of which 1171 (or one in 8.1) are Sophie Germain primes.
		Of these, 205 (or one in 5.7) are second-order Sophie Germain primes, and of these, 37 (or one in 5.5) are third-order Sophie Germain primes.
	www.canonical.org /~kragen/named-msgs/sophie-germain-primes (298 words)

	Biograpy of Germain
		Germain was not really interested in the study of mathematics until the age of 13 when the French Revolution started.
		Sophie Germain was 18, but she could not enroll in the academy because women were not allowed.
		A prime number must return prime answers in both equations, p and 2p+1, to be a Sophie Germain prime.
	www.andrews.edu /~calkins/math/biograph/199900/biogermn.htm (1483 words)

	The Largest Known Primes
		For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13 (the first 10,000, and other lists are available).
		The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.
		Altogether 37 of these primes are known, but since the region between the largest two and the previous primes has not been completely searched we do not know if the largest is 37th according to size.
	w3.impa.br /~gugu/mersenne/largest.html (1226 words)

	The Prime Glossary: Sophie Germain prime
		If both p and 2p+1 are prime, then p is a Sophie Germain prime.
		Around 1825 Sophie Germain proved that the first case of Fermat's last theorem is true for such primes.
		Soon after Legendre began to generalize this by showing the first case of FLT also holds for odd primes p such that kp+1 is prime, k=4, 8, 10, 14, and 16.
	primes.utm.edu /glossary/page.php?sort=SophieGermainPrime (175 words)

	RSA Coding
		Sophie Germain primes are still of interest today.
		We could use a Sophie Germain prime to obtain our two prime numbers, because when you have a Sophie Germain prime, you automatically get another prime number, by doubling it and adding one.
		In practice, as mentioned before, we start with prime numbers p and q that are at least 200 digits long.
	www.mathsci.appstate.edu /~sjg/class/1010/mathematician/rsa.html (1075 words)

	ipedia.com: Sophie Germain prime Article (Site not responding. Last check: 2007-11-06)
		They acquired significance because of Sophie Germain 's proof that Fermat's last theorem is true for such primes.
		A prime number p is called a Sophie Germain prime if 2p + 1 is also prime.
		A sequence {p, 2p + 1, 2(2p + 1) + 1,...} of Sophie Germain primes is called a Cunningham chain of the first kind.
	www.ipedia.com /sophie_germain_prime.html (220 words)

	PlanetMath: Germain prime
		It is conjectured that there are infinitely many Germain primes, but like the twin prime conjecture, this has not been proven.
		Cross-references: twin prime conjecture, twin prime constant, estimate, prime number
		This is version 2 of Germain prime, born on 2004-09-03, modified 2005-03-04.
	planetmath.org /encyclopedia/GermainPrime.html (71 words)

	The Top Twenty: Sophie Germain (p)
		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 63423 digits or more) plus twenty each of certain selected forms.
		The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, and 131.
		Heuristically, it seems reasonable to conjecture that there is a lower bound of this form as well.
	primes.utm.edu /top20/page.php?id=2 (284 words)

	Welcome to Prime Time
		As well as displaying ALL the time primes, the Time Prime clock also indicates whether the prime displayed is special, by illuminating its beacon in different colours.
		Thanks to Frank Wales, a helpful prime number enthusiast from the UK who pointed out a glaring typographical error in the website.
		Since there are an average of about three primes per minute -- you shouldn't blame the clock for being late.
	www.primetimeclock.com (306 words)

	TypeNumber.html
		If both a number (n) and 2n + 1 are prime, then n is a Sophie Germain Prime.
		a prime is regular if it does not divide the class number of the algebraic number field defined by adjoining a pth root of unity to the rationals.
		A Mersenne number 2^n-1 which is prime is called a Mersenne prime.
	www.adeptscience.co.uk /maplearticles/f223.html (802 words)

	Prime Definitions (Site not responding. Last check: 2007-11-06)
		Sophie Germain Primes - If a odd number, p, is prime for which 2p + 1 is also prime then it is said to be a Sophie Germain Prime.
		Twin Primes - A prime number, p, is said to be twin prime if p-2 or p+2 is also prime.
		3,5,7 are all twin primes but 23 is not a twin prime because 21 and 25 are not prime.
	www.bath.ac.uk /~ma2leb/pdefs.html (285 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Sophie Germain prime numbers are the result of her work in number theory.
		She proved that the first case of Fermat's last theorem is true for certain prime numbers - when both p and 2p + 1 are prime, then p is called a Sophie Germain prime, and the theorem is true.
		She sent one of her papers to the mathematician Lagrange who was so overwhelmed by it that he offered himself as her mentor.
	www.astr.ua.edu /4000WS/GERMAIN.html (237 words)

	[No title]
		Another way to make performance better for the client is not to use a Sophie Germain prime, but instead use one where p-1 has at least one large (160 bit) prime factor.
		The way to look for a Sophie Germain prime is to start with a number n, and then look for the smallest number p of the form 3 mod 8 larger than n where both p and (p-1)/2 are (probably) prime.
		If p passes this test, there is an overwhelming probability that p is a Sophie Germain prime and that 2 is a generator.
	www3.ietf.org /proceedings/01aug/I-D/draft-perlman-strong-cred-00.txt (3740 words)

	Maximally Periodic Reciprocals Bulletin of the Institute of Mathematics and its Applications 28 147-148 1992 (Site not responding. Last check: 2007-11-06)
		One simple yet excellent source of random numbers is the decimal expansion of reciprocals of the form 1/q, where q is a "suitable" prime number.
		Thus "suitable" prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to S = 3, 11, 23, 29, 83, 89, etc.).
		a 1812-digit Sophie Germain prime of the form 3 mod 20 discovered by Wilfred Keller (cited by Yates, S in "Tracking Titanics", a contribution to The Lighter Side of Mathematics (Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History), Guy, R. K., and Woodrow, R.E. eds.
	ourworld.compuserve.com /homepages/rajm/maximal.htm (274 words)

	More Exercises on Divisors of Integers (Site not responding. Last check: 2007-11-06)
		If p is prime and the natural numbers a and p have no common factor except 1, then a
		A Sophie Germain prime is an odd prime p for which 2p + 1 is also prime.
		Determine which of the prime numbers less than 20 are Sophie Germain primes.
	home.snu.edu /~lturner/MC-MathStr/IntegerDivisorsHW2.htm (200 words)

	"...And Superior Genius."
		These are links to sites dealing with the specific branches of mathematics in which Sophie concentrated her research throughout her lifetime.
		Palindromic Sophie Germain Primes: Harvey Dubner of Dubner Computing Systems has a site dedicated to discovering Sophie Germain Primes that read the same forwards and backwards...
		There is a whole section dedicated to Sophie and her role in the proof of this theorem.
	www.geocities.com /sophieg76/genius.html (309 words)

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