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	PlanetMath: Eisenstein prime
		Therefore no Mersenne prime is also an Eisenstein prime.
		Cross-references: Mersenne prime, primes, divisors, integers, Eisenstein integer, root of unity, complex
		This is version 3 of Eisenstein prime, born on 2006-08-15, modified 2006-08-21.
	planetmath.org /encyclopedia/EisenteinPrime.html (112 words)

	Sergei Eisenstein - Wikipedia, the free encyclopedia
		Eisenstein was a pioneer in the use of editing.
		Eisenstein completed a scenario by the start of October, 1930, with his two associates and the British author Ivor Montagu; but Paramount disliked it completely and, additionally, found themselves intimidated by the American fascist agitator Major Pease, who had mounted a public campaign against Eisenstein.
		Eisenstein suffered a hemorrhage and died at the age of 50.
	en.wikipedia.org /wiki/Sergei_Eisenstein (3315 words)

	Two
		Two is the smallest and the first prime number, and the only even one (for this reason it is sometimes humorously called "the oddest prime").
		It is also a Stern prime, a Pell number, and a Markov number, appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.
		Two is the base of the simplest numeral system in which natural numbers can be written concisely, being the length of the number a logarithm of the value of the number (whereas in base 1 the length of the number is the value of the number itself); the binary system is used in computers.
	www.brainyencyclopedia.com /encyclopedia/t/tw/two.html (2140 words)

	Eisenstein biography
		Eisenstein suffered all his life from bad health but at least he survived childhood which none of his five brothers and sisters succeeded in doing.
		Eisenstein died of pulmonary tuberculosis at the age of 29.
		Eisenstein, having laid the foundations for a theory of elliptic functions, was able to carry out much of his design for the building itself, and to indicate how he wished it completed.
	www-history.mcs.st-and.ac.uk /history/Biographies/Eisenstein.html (2448 words)

	Eisenstein
		Eisenstein worked on a variety of topics including quadratic and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws.
		From 1846 to 1847 Eisenstein worked on elliptic functions and in the first of these years he was involved in a priority dispute with Jacobi.
		Eisenstein spent a year in Sicily in an attempt to improve his health but after his return to Germany he died of pulmonary tuberculosis at the age of 29.
	library.wolfram.com /examples/quintic/people/Eisenstein.html (318 words)

	Add new comment \| Linux Journal
		Primes are a fundamental part of our numbering system, and the search for prime numbers has fascinated mathematicians for more than two millennia.
		Prime Pages not only archives the world's largest primes, but it also is the world's most complete resource for information on prime numbers.
		Trying to figure out how many primes are in a range and what the distribution looks like within that range is an active area of research that helps drive the search for prime numbers.
	www.linuxjournal.com /comment/reply/8096 (2596 words)

	167 (number) Information
		It is a prime number, a safe prime and an Eisenstein prime with no imaginary part and a real part of the form 3n − 1.
		Since the next odd number, 169, is a square of a prime, 167 is a Chen prime.
		It is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits.
	www.bookrags.com /wiki/167_(number) (145 words)

	Ferdinand Eisenstein Summary
		Eisenstein was inspired to study mathematics after meeting William Rowan Hamilton, who gave him a copy of a paper he had recently written on a difficult problem in mathematics.
		Like Galois and Abel, Eisenstein died before the age of 30, and like Abel, his death was due to tuberculosis.
		Gauss's choice of Eisenstein, who specialized in number theory and analysis, may seem puzzling to many, but it is justified by the fact that Eisenstein easily proved several results that were unattainable even for Gauss, like the theorem on biquadratic reciprocity.
	www.bookrags.com /Ferdinand_Eisenstein (275 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Then if f is Eisenstein at p, then p is totally ramified in the extension Q(a) of Q. There are examples of number fields no prime in which is totally ramified.
		I'm assuming that the converse of Eisenstein's Criterion for irreducible.
		This f(x) is not Eisenstein with respect to any prime p, and you can't get an Eisenstein polynomial with respect to any prime p FROM it, by any "reasonable" transformation of the roots.
	www.math.niu.edu /~rusin/known-math/99/eisenstein (612 words)

	4-dim HyperDiamond Lattice
		At the moment the largest known Mersenne prime is 2^13466917 - 1 (which is also the largest known prime) and the corresponding largest known perfect number is 2^13466916(2^13466917 - 1).
		Here the primes are: sqrt(5) and its associates; rational primes 5n +/- 2 and their associates; factors a + bi of rational primes 5n +/- 1 Hardy and Wright use such algebraic extensions, by sqrt(5) and sqrt(3), to prove primality of some Mersenne primes.
		Conway and Sloane use algebraic extension of quaternions by the Golden (1/2)(1 + sqrt(5)) to construct the 8-dimensional E8 lattice and the 24-dimensional Leech lattice.
	www.valdostamuseum.org /hamsmith/PrimeFC.html (5922 words)

	Irreducibility Criteria
		Notice that Eisenstein's criterion essentially reduces the problem of factoring a polynomial of degree d to a problem of factoring d integers, the coefficients of the transformed polynomial, to see if they share a suitable common prime divisor.
		Thus, the maximum number of times that g(k) could equal either +1 or -1 is 2(d_g), and by the same reasoning the max number of times that h(k) could equal +-1 is 2(d_h).
		Therefore, the maximum number of integers k differing by more than 2 for which g(k)=+-1 is only d_g, and the maximum number of such integers for which h(k)=+-1 is only d_h, for a combined total of (d_g + d_h) = d.
	www.mathpages.com /home/kmath406.htm (917 words)

	2 (NUMBER) FACTS AND INFORMATION
		Two is the smallest and the first prime number, and the only even one.
		It is also a Stern prime, a Pell number, and a Markov number, appearing in infinitely many solutions to the Markov Diophantine equation involving odd-indexed Pell numbers.
		Despite being a prime, two is also a highly composite number, because it has more divisors than one.
	www.abusinessforme.com /2_(number) (1376 words)

	[No title]
		It is fortuitous that the Eisenstein integers are a PID, because it means that irreducible numbers (n = ab ==> na or nb) are prime (nab ==> na or nb).
		Therefore 5, 11, 17, etc. are Eisenstein primes, but 7, 13, 19, etc. are not.
		Finally, 3 is not an Eisenstein prime, but since has only one Eisenstein prime factor that appears twice, rather than two distinct ones, its presence complicates the set of solutions but it does not produce solutions ex nihilo.
	www.math.niu.edu /~rusin/known-math/93_back/calc.int (889 words)

	Amazon.com: The Crystal Palace: Books: Phyllis Eisenstein (Site not responding. Last check: 2007-10-10)
		I have nothing wrong with character development, but I would have preffered that Eisenstein had drawn out a story in which the character development could happen "on the move" as it did in the first book.
		It seemed like Eisenstein kept on trying to add on special evil powers to him all through her book to cover what she realised was a rather weak archfiend.
		Eisenstein is a mistress of her art and this book serves to exemplify her skill.
	www.amazon.com /Crystal-Palace-Phyllis-Eisenstein/dp/0451156781 (1040 words)

	227 (number) - TvWiki, the free encyclopedia (Site not responding. Last check: 2007-10-10)
		227 is a prime number, and a twin prime with 229 (thus 227 is a Chen prime).
		Since (227 - 1)/2 is 113, also a prime, 227 is a safe prime.
		227 is a Stern prime since it is not of the form $p + 2b^2$ .
	www.tvwiki.tv /wiki/227_(number) (129 words)

	Irreducibility Question (Rationals, Eisenstein!) Text - Physics Forums Library
		(Q being the rational fancy Q) So I used Eisenstein Criterion (because we are dealing with the rationals) and said that the coefficents are: 1 (for x^4), 2(for x^2), 2 with the last 2 coefficients being the important ones.
		Suppose that there is some prime p such that pa_{1}, pa_{2},..., pa_{n}, but p^2 does not divide a_{n} Then f(x) is irreducible in Q[x].
		I just looked in another book, and on wikipedia, at the definition of The Eisenstein Criterion and they both mention a property about the first coefficient (1 for x^4); specifically: p cannot divide the first coefficient(1 for x^4).
	www.physicsforums.com /archive/index.php/t-102315.html (412 words)

	blog.myspace.com/theunabeefer
		Forty-seven is the 15th prime number, a safe prime, a supersingular prime, and the 6th Lucas prime.
		It is an Eisenstein prime with no imaginary part and real part of the form 3n - 1.
		Its representation in binary being 101111, 47 is a prime Thabit number, and as such is related to the pair of amicable numbers {17296, 18416}.
	blog.myspace.com /index.cfm?fuseaction=blog.view&friendID=3756325&blogID=182224669 (972 words)

	Homepage of Matti Pitkänen
		, k prime, are k=151, 157, 163, 167 and correspond to the p-adic length scales 10 nm, 80 nm, 640 nm, and 2560 nm.
		In biologically interesting length scales four nearby Gaussian primes corresponding to k=151,157,163,167 appear and correspond to G-adic length scales L(k) varying from the thickness of the cell membrane to bacterial size (256 μm).
		The requirement that genetic code, which corresponds to small prime p=127, is realized in DNA length scales, suggests strongly that small-p p-adicity is realized in biological length and time scales.
	www.physics.helsinki.fi /~matpitka/cbookI/newcbookI2001.html (6498 words)

	11th - Trade Encyclopedia (Site not responding. Last check: 2007-10-10)
		11 is the fourth Sophie Germain prime, the third safe prime, and the first repunit prime.
		The next prime is 13, with which it comprises a twin prime.
		Because it has a reciprocal of unique period length among primes, 11 is the second unique prime.
	www.bestbuy.tiptophot.com /trade/index.php?title=11th (765 words)

	P.I.E.S. - Prime Internet Eisenstein Search
		PIES is a new project (2003/08/29) interested both in investigating the properties of Generalised Eisenstein Fermat numbers, or GEFs, and in finding large prime GEFs.
		It is Eisenstein who gives his name to the Eisenstein Integers, and it is these points on an equilateral triangular lattice in the complex plane which interest us.
		For efficiency, I intend to sieve the full testable range for the highest exponents to a depth of ~1000T, a depth rarely reached in the throes of prime hunting, except for, of course, GFN sieving.
	fatphil.org /maths/PIES/faq.html (738 words)

	Math Forum Discussions
		Proof of the Infinitude of Fermat primes 2^(2^n) +1 Re: Proof of the Infinitude of Twin Primes, using the method of Euclid
		Re: Proof of the Infinitude of Fermat primes 2^(2^n) +1 Re: Proof of the Infinitude of Twin Primes, using the method of Euclid
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?messageID=3983635&tstart=0 (837 words)

	pepperknit » Blog Archive » 29 (Site not responding. Last check: 2007-10-10)
		29 is the tenth prime number, and also the third primorial prime, the next prime number as well as primorial prime being thirty-one, with which it comprises a twin prime.
		29 is a Lucas prime, a Pell prime, and a tetranacci number.
		29 is an Eisenstein prime with no imaginary part and real part of the form 3n − 1.
	pepperknit.com /blog/archives/301 (668 words)

	Titan Biographies: Prime Internet Eisenstein Search
		A titan, as defined by Samuel Yates, is anyone who has found a titanic prime.
		This page provides data on Prime Internet Eisenstein Search, one of the projects which have found these primes.
		These are to the Eisenstein integers Z(ω) what Generalised Fermat Numbers are to the Rational and Gaussian Integers.
	primes.utm.edu /bios/page.php?id=541 (224 words)

	179 (number) at AllExperts
		179 is a prime number, and both a Sophie Germain prime and a safe prime, being the second term in the smallest Cunningham chain of the first kind that is six terms long, preceded by 89 and followed by 359.
		179 is a twin prime with 181, making 179 a Chen prime.
		Since 179 is one less than a multiple of three, it is an Eisenstein prime with no imaginary part.
	en.allexperts.com /e/0/179_(number).htm (197 words)

	Errata and remarks for Gauss and Jacobi sums, by Berndt, Evans, Williams
		is a primary Eisenstein prime whose norm p is congruent to 1 modulo 3, and
		We can give a short proof of this as follows, using what was already proved so far.
		For example, r = 7 is a cubic residue of the prime p = 73.
	math.ucsd.edu /~revans/errata/errata.html (452 words)

	Homepage of Matti Pitkänen
		Gaussian Mersennes are Gaussian primes of the form u^k-1, u=1+i or u=1-i.
		In biologically interesting length scales four nearby Gaussian primes corresponding to k=151,157,163,167 appear and correspond to G-adic length scales L(k) varying from the thickness of the cell membrane to bacterial size (256 micrometers).
		Infinite primes are a crucial element of TGD inspired theory of consciousness.
	www.helsinki.fi /~matpitka/newcbook.html (5830 words)

	Eisenstein's Proof
		The Quadratic Reciprocity Theorem compares the quadratic character of two primes with respect to each other.
		Here is Eisenstein's proof, closely following both his own language and notation (which he conveniently and successfully abuses).
		Eisenstein now uses a geometric representation of the exponent in this last equation to transform it twice while retaining its parity: This exponent is precisely the number of integer lattice points with even abscissas lying in the interior of triangle ABD in the Figure (note that no lattice points lie on the line AB).
	www.math.nmsu.edu /~history/eisenstein/node2bak.html (475 words)

	Internet-based Distributed Computing Projects - Updates
		This is the 4th-largest (and largest non-Mersenne) prime number known.
		GIMPS and Michael Shafer discovered the 40th known Mersenne prime, 2^20,996,011 - 1 (6.3 million digits) on November 17, 2003.
		It is the largest known prime number, and the 6th Mersenne prime found by the project.
	www.distributedcomputing.info /distrib-2003/updates.html (791 words)

	Titan Biographies: Phil Carmody's ForEis
		This page provides data on Phil Carmody's ForEis, one of the programs which have found these primes.
		A Probable Primality checker for numbers of the form Phi(2^s3^t,b), the Generalised Eisenstein* Fermats.
		Numbers of the form Phi(2^s*3,b) are exceptionally quick (compared to other programs capable of PRPing such forms) due to the use of a bizarre and novel transform algorithm.
	primes.utm.edu /bios/page.php?id=543 (208 words)

	blog.myspace.com/krhaydon
		Numerical analysis aside, I was experiencing a mild sensation - resembling pride - at the thought of being an Einstein Prime.
		On a second look, I realize I am no such thing...no physical manifestation of tolerance and mystery...no brain child of the world's most famous tongue-sticker-outer.
		Eisenstein, who's alien gibberish caused a sensation amongst the alienated and gibberating mathematical geniuses of 19th century Germany.
	blog.myspace.com /krhaydon (1206 words)

	The On-Line Encyclopedia of Integer Sequences
		Primes p dividing sum(k=0,p,C(2k,k)) -1 = A006134(p)-1 - Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 08 2003
		A. Granville and G. Martin, Prime number races
		These are the primes arising in A024893, A087370, A088879, A091177 gives prime index.
	www.research.att.com /~njas/sequences/A003627 (122 words)

	Resources for "The Prime Internet Eisenstein Search" (Site not responding. Last check: 2007-10-10)
		Prime numbers: a computational perspective, Springer-Verlag, New York, 2001.
		The new book of prime number records, 3rd edition, Springer-Verlag, New York, 1995.
		Prime numbers and computer methods for factorization, Birkhauser, Boston, 1994.
	www.linuxjournal.com /node/8273/print (67 words)

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