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Topic: Germain primes

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	Sophie Germain
		She was born to a middle-class merchant family in Paris, France, and began studying mathematics at age 13, despite strong attempts to dissuade her from engaging in a 'men's profession' by her parents.
		Germain was particularly interested in Joseph-Louis Lagrange's teachings and submitted papers and assignments under the pseudonym Monsieur Le Blanc, a former student of Lagrange's.
		In 1811 Germain entered The French Academy of Sciences[?]' contest to explain the underlying mathematical law of a German mathematician, attempting to explain Ernst Chladni's study on vibrations[?] of elastic surfaces.
	www.ebroadcast.com.au /lookup/encyclopedia/so/Sophie_Germain.html (558 words)

	PlanetMath: Sophie Germain
		Germain made headway towards proving Fermat's last theorem.
		Today she is best known for the Sophie Germain primes.
		This is version 2 of Sophie Germain, born on 2006-10-02, modified 2006-10-08.
	planetmath.org /encyclopedia/SophieGermain.html (215 words)

	Sophie Germain prime - Wikipedia, the free encyclopedia
		A Sophie Germain prime p > 3 is of the form 6k - 1 or, equivalently, p ≡ 5 (mod 6).
		Sophie Germain primes were the subject of the eponymous proof in the stage play Proof and the subsequent film Proof.
		Sophie Germain primes have a practical application in the generation of pseudo-random numbers.
	en.wikipedia.org /wiki/Sophie_Germain_prime (536 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 Germain
		Germain never married and was funded by her father, while spending the remainder of the revolution studying differential calculus.
		Germain's approach was more general and consisted a goal of not proving that one particular equation had no solutions, but to state something about several of them.
		Sophie Germain was diagnosed with breast cancer and fought the disease for most of her life.
	www.andrews.edu /~calkins/math/biograph/biogermn.htm (1689 words)

	type
		Primes where all of the digits have holes; unholey primes do not have any digits with holes in them.
		A prime that must have a total even number of digits, and the digits in the first half of the prime must digger from the corrseponding digits of the second half, resulting in a coincidence ratio of 0.
		Primes that are named because they are usually expressed in their subscriptal notation.
	www.fortunecity.com /meltingpot/manchaca/799/type.html (541 words)

	[No title]
		If there are only a finite number of primes, then the right side will not diverge, but we already know that the left side will, so we have a contradiction, so there must be an infinite number of primes.
		A pair of Sophie Germain primes is a set of two prime numbers p and 2p+1.
		Sophie Germain was the first person to attempt a general proof of Fermat's Last Theorem, rather than proving it for individual prime exponents.
	www.albanyconsort.com /primes/primes.html (819 words)

	Primitive Roots and Exponential Iterations
		Incidentally, a prime q such that 2q+1 is also a prime is known as a “Germain prime”. These were first studied by Sophie Germain in relation to the "first case" of Fermat’s Last Theorem. From Proposition 1 we see that every primitive exponential mapping corresponds to a Germain prime.
		Empirically such primes are fairly common, but it has never been proven that there are infinitely many such primes.
		Since this is a prime, and since (m – 1)/2 = 131 is also a prime, we know primitive exponents modulo 263 exist, and they are the primitive roots modulo 131.
	www.mathpages.com /home/kmath148/kmath148.htm (2076 words)

	Sophie Germain and FLT
		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.
		In 1951, P Dénes extended Germain and Legendre's original result by proving that if n is an odd prime and p=2kn+1 is a prime, where k is not a multiple of 3 and k < 54, then the first case of Fermat's Last is true for the exponent n.
		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)

	On Case 1 of Fermat's Last Theorem
		Let's call the prime p a "Germain Prime" relative to the positive integer n if there exists an integer k such that p=kn+1 and the equation a+b=c (mod p) has no solutions with a,b,c each a kth root of 1 (mod p).
		For example, relative to n=2 the only Germain Primes are 3 and 5, so it follows that for every Pythagorean triple x,y,z we have xyz divisible by 3 and 5.
		Following is a table of all the Germain Primes with k < 400 for integers n (not necessarily primes as in the above table) from 1 to 13.
	www.mathpages.com /home/kmath367.htm (1792 words)

	PlanetMath: Sophie Germain prime (Site not responding. Last check: 2007-09-10)
		It is conjectured that there are infinitely many Sophie Germain primes, but (like the Twin Prime Conjecture) this has not been proven.
		Cross-references: twin prime constant, twin prime conjecture, prime number
		This is version 5 of Sophie Germain prime, born on 2004-09-03, modified 2006-09-01.
	planetmath.org /encyclopedia/GermainPrime.html (72 words)

	Sophie Germain (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		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 Germain died at the age of 55, on June 27, 1831, after a battle with breast cancer.
	www.agnesscott.edu.cob-web.org:8888 /lriddle/women/germain.htm (1552 words)

	Sophie Germain
		Even today, it is felt that she was never given as much credit as she was due for the contributions she made in number theory and mathematical physics because she was a woman.
		Germain's theorem is a major step toward proving Fermat's last theorem for the case where n equals 5" (Dalmedico 119).
		The street Rue Sophie Germain in Paris has been named in her honor and a statue of her now stands in the courtyard of the Ecole Sophie Germain, also in Paris.
	www.agnesscott.edu /lriddle/women/germain.htm (1552 words)

	RSA Key generation
		For a large number X, the number of primes less than X is about X/ln(X); so the probability of a number near X being prime is roughly 1/ln(X).
		Sophie Germain primes are primes P such that 2P+1 is also a prime; CTClib's Sophie Germain prime generation method for RSA keypairs produces a key pair whose primes are of the form 2P+1 for some smaller prime.
		Some speed-up of primality testing is possible, to reduce the number of large integer operations performed in the initial checking against small prime factors though it would take some significant reworking of the code.
	www.ravnaandtines.com /rsa.html (424 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		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)

	primes
		− 1 to be prime, q must be prime, but not all numbers of this form are prime.
		Mersenne primes are the subject of the Great Internet Mersenne Prime Search, with a $100,000 prize for the first to discover a prime with more than 10 million decimal digits.
		A pair of primes such that one is twice the other plus 1, for example "11" and "23".
	www.sizes.com /numbers/primes.htm (297 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).
		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.
		A Sophie Germain prime is an odd prime p for which 2p+1 is also a prime.
	w3.impa.br /~gugu/mersenne/largest.html (1226 words)

	Sophie Germain (Site not responding. Last check: 2007-09-10)
		It is through this work that we have a special group of prime numbers known a s Germain primes.
		Germain primes are those prime numbers p such that 2p +1 is also a prime number.
		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)

	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 Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors.
		Because the way the largest numbers N are proven prime is based on the factorizations of either N+1 or N-1, and for Mersennes the factorization of N+1 is as trivial as possible (a power of two).
	www.utm.edu /research/primes/largest.html (1152 words)

	[No title]
		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.
		Twin primes are primes of the form p and p+2, i.e., they differ by two.
		These were named after Sophie Germain when she proved that the first case of Fermat's Last Theorem (xn+yn=zn has no solutions in non-zero integers for n>2) for exponents divisible by such primes.
	w3.impa.br /~gugu/mersenne/largest.txt (1308 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 77429 digits or more) plus twenty each of certain selected forms.
		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 /top20/page.php?id=2 (308 words)

	Euler and other mathematicians' contribution
		The most important of all is Sophie Germain’s Theorem which directly led to the proof of the case n=5 given by Dirichlet and Legendre.
		The approach intends to proof that for n=p where p is a prime and 2p+1 is also a prime(p is called Germain Primes), then one of the three solutions, x, y or z must be divisible by n.
		Through this theorem, any proofs for cases that n= Germain Primes only need to show that for all solutions x, y and z are not divisible by n.
	library.thinkquest.org /28049/Euler.htm (755 words)

	South Coast Repertory Playgoers Guide - 'Proof'
		The smallest prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41.
		One of her most notable accomplishments is the discovery of a new set of prime numbers which are now known as Germain primes.
		For example, 2 is prime, 2 double equals four, plus one equals five, which is also a prime.
	www.scr.org /season/02-03season/studyguides/proof/glossary.html (977 words)

	D Numbers
		For every prime p it is easy to find many primes q compatible with p such that pq is in D. Here is a text file of the first 5000 primes* p and its first such q: (pq.txt).
		First 300 primes p and the first q such that p^2*q is in D (p2q.txt).
		The primes p=17, 23, 47, 53, and 83 all have the property observed with 3p^2, namely that each of the sets S(p,j), j=1,2,3,4, all contain an element divisible by 5 and the set S(p,5) has a composite so pq^2 is not in D for any q.
	glory.gc.maricopa.edu /~wkehowsk/propertyd/index.html (1379 words)

	Biograpy of Sophie Germain
		The last paper she wrote was an outline to a philosophical essay that was published posthumously as "Considérations générale sur l'état des sciences et des lettres" in the Oeuvres philosophiques.
		Sophie Germain was diagnosed with breast cancer and fought over it for most of her life.
		P is an odd Sophie Germain prime, then there do not exist intergers x, y, and z different from 0 and not multiples of p such that x
	www.andrews.edu /~calkins/math/biograph/199899/biogermn.htm (1272 words)

	nrich.maths.org::Mathematics Enrichment::NRICH
		As for the last question, I think I remember someone at college saying that a Germain prime n is one such that n,2n+1 are prime.
		We will drop the condition that n is prime and only insist n+1 is a multiple of 4 not 12.
		Let p be an odd prime, and let a be natural number which isn't divisible by p.
	nrich.maths.org /askedNRICH/edited/2186.html (519 words)

	PalPrimePage 2
		Despite having a length that is not a prime nor a palindrome
		Primes were verified using the APRT-CLE program in UBASIC.
		P, Q, R all have 727 digits, the number of digits is a palindromic prime.
	users.skynet.be /worldofnumbers/palprim2.htm (1093 words)

	49. Sierpinski-like numbers
		However, my computations showed that 1472^n+/-1 are both prime* for n=44, and that 2132^n+/-1 are both prime* for n=36, so h=147 and h=213 may be removed from the list.
		However, adding three primes to the given example, I was able to construct a covering using 19 primes, which are all divisors of 2^720 - 1.
		of P are just the prime components of the covering set, while the additional factor 2 ensures that every other k is also odd, and the factor 3 ensures that every k is also a multiple of 3.
	www.primepuzzles.net /problems/prob_049.htm (2402 words)

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