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Topic: Sieve theory

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Number theory

Twin prime conjecture

General number field sieve

Sieve of Eratosthenes

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	Sieve theory - Wikipedia, the free encyclopedia
		Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers.
		One of the original purposes of sieve theory was to try to prove conjectures in number theory such as the twin prime conjecture.
		Sieve theory is somewhat related to sieve algorithms such as the general number field sieve, used for factoring large numbers, although sieve theory and sieve algorithms serve different purposes.
	en.wikipedia.org /wiki/Sieve_method (587 words)

	Sieve - Wikipedia, the free encyclopedia
		A sieve is a device for segregating aggregate based on particle size.
		Sieves commonly consist of a wire mesh on which the aggregate is placed.
		A sieve used in cooking is a colander.
	en.wikipedia.org /wiki/Sieve (201 words)

	PlanetMath: Brun's pure sieve
		The sieve of Eratosthenes is based on the principle of inclusion-exclusion in the form
		This inequality is stronger than the one obtained by an application of the sieve of Eratosthenes.
		This is version 6 of Brun's pure sieve, born on 2004-02-21, modified 2004-07-31.
	planetmath.org /encyclopedia/BrunsPureSieve.html (552 words)

	Sieve of Eratosthenes - Wikipedia, the free encyclopedia
		In mathematics, the Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer.
		It is the predecessor to the modern Sieve of Atkin, which is faster but more complex.
		Wheel factorization is often applied on the list of integers to be checked for primality, before Sieve of Eratosthenes is used, to increase the speed.
	en.wikipedia.org /wiki/Sieve_of_Eratosthenes (300 words)

	Introduction to Arithmetic: Number Theory; Prime Numbers, Fermat Theorem, Goldbach Conjecture and Diophantine Equations (Site not responding. Last check: 2007-11-06)
		Number theory is the branch of mathematics concerned with studying the properties and relations of integers.
		Algebraic number theory is the study of numbers that are the roots of polynomial equations with integer coefficients, and includes the study of Gaussian integers.
		The Sieve of Eratosthenes is a routine for compiling a table of prime numbers devised by the Greek mathematician Eratosthenes in the 3rd century BC.
	www.geocities.com /mathfair2002/school/arit/arithm3.htm (2234 words)

	UIUC Number Theory: Faculty Research Descriptions (Site not responding. Last check: 2007-11-06)
		His main area of work is multiplicative number theory, particularly elementary proofs of the prime number theorem, the theory of Beurling generalized numbers, and sieve theory.
		The object of sieve theory is to estimate the number of elements that remain in a set if those satisfying certain congruence conditions are removed.
		She is currently interested in integrable models using the algebraic approach; representations of quantum affine algebras; and combinatorial representation theory of vertex operator algebras and conformal field theories.
	www.math.uiuc.edu /ResearchAreas/numbertheory/facultyresearch.html (3443 words)

	Twin prime conjecture - Wikipedia, the free encyclopedia
		The twin prime conjecture is a famous problem in number theory that involves prime numbers.
		This famous result was the first use of the Brun sieve and helped initiate the development of modern sieve theory.
		The approach he took involved a topic called sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjecture in similar manners.
	en.wikipedia.org /wiki/Twin_Prime_Conjecture (597 words)

	Encyclopedia: Sieve
		In mathematics, the sieve of Eratosthenes is a simple algorithm for finding all the prime numbers up to a specified integer.
		In mathematics, the general number field sieve is the most efficient algorithm known for factoring integers.
		In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain.
	www.nationmaster.com /encyclopedia/Sieve (431 words)

	Debug Strategy (Site not responding. Last check: 2007-11-06)
		During the first iteration the theories have to be based on the external observations and simple conclusions from the source code.
		Truth of all predictions made by a theory does not in itself prove the theory to be correct.
		There are three families of theories, it was always there, it appeared during the first loop, and it appeared during the second loop.
	home.earthlink.net /~patricia_shanahan/debug (4883 words)

	Math 571 Analytic Number Theory I
		Modern forms of the sieve are due to Brun (1885-1978) and Selberg.
		Fundamental to many of the analytic methods in number theory are questions as to how closely a given real number can be approximated by a rational number with denominator not exceeding a given quantity, and generalisations of this are related to Minkowski's theorem in the geometry of numbers.
		The large sieve, Selberg's lambda-squared sieve and Gallagher's sieve.
	www.math.psu.edu /rvaughan/Math571F04.html (463 words)

	11: Number theory
		Number theory is one of the oldest branches of pure mathematics, and one of the largest.
		For example, "additive number theory" asks about ways of expressing an integer N as a sum of integers a_i in a set A. If we set f(z) = Sum exp(2 pi i a_i z), then f(z)^k has exp(2 pi i N z) as a summand iff N is a sum of k of the a_i.
		Questions in algebraic number theory often require tools of Galois theory; that material is mostly a part of 12: Field theory (particularly the subject of field extensions).
	www.math.niu.edu /~rusin/known-math/index/11-XX.html (2537 words)

	module 3 notes
		This is known as the cohesion tension theory of water transport.
		Sieve tube contents continue to exude from stylet, can be collected and analysed.
		The reason aphids are used is because normally when a sieve tube is punctured with a measuring probe, the holes in its end walls quickly plug up.
	www.mrothery.co.uk /plants/planttransportnotes.htm (1446 words)

	Goldbach's conjecture - One Language (Site not responding. Last check: 2007-11-06)
		In mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics.
		This result was subsequently improved by many authors; currently, the best known result is due to Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most seven primes.
		Chen Jingrun showed in 1966 using the methods of sieve theory that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes)—e.g., 100 = 23 + 7·11.
	www.onelang.com /encyclopedia/index.php/Goldbach%27s_conjecture (1334 words)

	General Theory
		Applying the Nash sieve to a fixed range of n to eliminate from consideration any value covered by (at least) one of the primes leaves a list of n which either have to be tested for primality directly or subjected to a larger sieve.
		This sieve is used to search for numbers of high Nash weight.
		Basically, both a positive and a negative sieve are applied to the value of k and a Nash weight obtained by counting the number of exponents not covered by either sieve.
	www.glasgowg43.freeserve.co.uk /nashdef.htm (2448 words)

	Computational number theory
		Computational number theory studies problems from elementary, algebraic and analytic number theory which require the help of fast computers, particularly vector and parallel systems.
		The purpose of this research is to contribute to the development of the algebraic theory of discrete tomography (DT) and to write computer programs which are based on the developed theory and are aimed at practical applications, e.g.
		In particular, the generation of proper polynomials used by the siever, the effect of using three large primes in the relations generated by the siever, and the possibility to improve the efficiency of the linear algebra step in NFS will be studied.
	db.cwi.nl /projecten/project.php4?prjnr=84 (1003 words)

	Kids.net.au - Encyclopedia Twin prime conjecture -
		The twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers.
		In 1966, Jing-run Chen showed that there are infinitely many primes p such that p +2 is a product of at most two prime factors.
		There is also a generalization of the Twin Prime Conjecture, known as the Hardy - Littlewood conjecture, which is concerned with the distribution of twin primes, in analogy to the prime number theorem.
	www.kids.net.au /encyclopedia-wiki/tw/Twin_Prime_Conjecture (378 words)

	NFSNET
		The number field sieve (NFS) is the asymptotically fastest known algorithm for factoring large composite integers which have no known special form.
		The number field sieve is particularly efficient for numbers of the form: N=r^e+s where r and s are small.
		Even a basic description of the number field sieve is beyond the scope of this FAQ however there are several books that provide excellent descriptions CFRAC and the QS and go further to explain at least the basics of the number field sieve.
	www.nfsnet.org /faq-nfs.html (510 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The first part will explain how the Sieve of Eratosthenes, a simple method of computing the primes up to N, can be sped up to O(N) additions with roughly N^{1/2} bits of storage, or O(N/log log N) additions with roughly N bits of storage.
		Estimating factorization time.'' This was a talk on estimating the speed of the quadratic sieve and the number field sieve.
		This talk will give examples of sieving, explain the relevance of sieving to factorization, and describe a sieving circuit that is asymptotically much faster than previously published hardware designs at the same cost.
	crypto.sericyb.com.au /talks.html (3033 words)

	The Prime Glossary: sieve of Eratosthenes
		The next number left, 7, is larger than the square root of 30, so there are no multiples of 7 to cross off that haven't already been crossed off (14 and 28 by 2, and 21 by 3), and therefore the sieve is complete.
		In fact the problem with the algorithm as presented above is not really speed (it uses O ( n ( log n)log log n) bit operations), but rather space (it is O( n)).
		Hudson, "The segmented sieve of Eratosthenes and primes in arithmetic progression," Nordisk Tidskr.
	primes.utm.edu /glossary/page.php?sort=SieveOfEratosthenes (593 words)

	CanadaRED Directory : Science : Math : Number Theory : Prime Numbers (Site not responding. Last check: 2007-11-06)
		An interactive animation of the sieve of Eratosthenes to obtain prime numbers (JavaScript required).
		An introductory page on the theory of prime numbers with proofs of some important theorems including the infinitude of primes, Euclid's algorithm and the fundamental theorem of arithmetic.
		Research in number theory, prime numbers and parity of arithmetic functions.
	canadared.com /index.php?c=Science/Math/Number_Theory/Prime_Numbers (972 words)

	Ancient Greeks: Prime Numbers and Number Theory
		The proof was so powerful that it caused a crisis to the Pythagorean theory of numbers.
		These books contain a treatment of the Euclidean algorithm, a proof that there are infinitely many primes, a theory of incommensurable (or irrationals -- the original proof that the square root of 2 is irrational is attributed to Pythagoras).
		Thus he kept 2 but threw out 4, 6, 8, etc. His sieve then kept 3 and rejected 6, 9, 12, etc. This method is very efficient for numbers of up to 1000000.
	www.mlahanas.de /Greeks/Primes.htm (2713 words)

	FSTTCS 2002: Workshop on Number Theory (Site not responding. Last check: 2007-11-06)
		A Workshop on Coding and Number Theory, will be held on the two days after the FST-TCS conference at IIT, Kanpur, India, December 15-16.
		Sieve theory is very useful in counting numbers with the given properties.
		This tutorial will introduce the basics of Sieve theory and enumerate the main techniques used in such countings.
	www.fsttcs.org /2002/workshop_nmbrthry.html (230 words)

	Tecnoseek Science : Math : Number Theory : Prime Numbers (Site not responding. Last check: 2007-11-06)
		Animation of Eratosthenes' Prime Sieve - An interactive animation of the sieve of Eratosthenes to obtain prime numbers (JavaScript required)..
		Basic Theorems Concerning Prime Numbers - An introductory page on the theory of prime numbers with proofs of some important theorems including the infinitude of primes, Euclid's algorithm and the fundamental theorem of arithmetic..
		Introductory Prime Number Theory Resources - Notes and links compiled by Mark Watkins on the relation of the Riemann zeta function to the distribution of prime numbers..
	www.tecnoseek.it /Dir/Science/Math/Number_Theory/Prime_Numbers (1352 words)

	A Survey on the Number Field Sieve (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		The Number Field Sieve is expected to be the fastest among the recent major factoring methods.
		Keywords: Factoring Method, Sieving Method, NFS, SNFS, GNFS, DLP Recent developments of computational number theory have been very much infuenced by cryptography.
		In this note, a survey on the Number Field Sieve, or NFS, will be given as one of the most important of these methods.
	citeseer.ist.psu.edu /312752.html (547 words)

	Analytic Number Theory course/college Analytische Getaltheorie, Leiden
		The Mathematical atlas, in particular the subcategories Number Theory (2000 MSC 11), Zeta and L-functions: analytic theory (2000 MSC 11M) and Multiplicative number theory (2000 MSC 11N).
		Later, several other proofs of the Prime Number Theorem were given, all based on complex analysis, until in 1948 Erdös and Selberg independently published an "elementary proof" of the Prime Number Theorem, avoiding complex analysis.
		Dirichlet may be viewed as the founder of analytic number theory.
	www.math.leidenuniv.nl /~evertse/ant.shtml (1207 words)

	Number Theory - The Science Beat ... Scientific Research, Space, NASA - SearchBeat.com (Site not responding. Last check: 2007-11-06)
		Number Theory Foundation - Aims to collect donations from supporters of number theory and to disburse these donations to encourage research.
		Computational Projects in Number Theory - Interesting problems, usually requiring extensive verifications or enumerations, to occupy the idle CPU time of workstations and personal computers.
		The Valuation Theory Home Page - A forum for all mathematicians who work in valuation theory or apply valuation theoretical results in their own field of research.
	www.searchbeat.com /Science/Math/NumberTheory (954 words)

	Asymptotic Theory for Sieve Estimators in Semimartingale Regression Models - Storming Media
		McKeague (1986) introduced sieve estimators for alpha sub 1,..., alpha sub p and established consistency in sq l-norm.
		In this paper the asymptotic distribution theory for the integrated sieve estimators is developed.
		Smoothed sieve estimators are shown to be pointwise consistent and rates of convergence are provided.
	www.stormingmedia.us /70/7029/A702961.html (178 words)

	1.1. Number Theory (Site not responding. Last check: 2007-11-06)
		, B. A world wide number field sieve factoring record: on to 512 bits, proceedings of the Asiacrypt conference, held in Kyongju, South Korea, November 1996 and published as K.
		R.-M. A multiple polynomial Number Field Sieve, proceedings of the ANTS-II conference, held in Talence, France, May 1996 and published as H.
		R.-M. The number field sieve on many computers, in: Proceedings of the Fifth Conference of the Canadian Number Theory Association and will be published as R.
	math.leidenuniv.nl /~stieltjes/archief/biennial9596/frame/node74.html (994 words)

	Prime Numbers : Science : Math : Number Theory : Prime Numbers (Site not responding. Last check: 2007-11-06)
		Animation of Eratosthenes' Prime Sieve : An interactive animation of the sieve of Eratosthenes to obtain prime numbers (JavaScript required).
		Basic Theorems Concerning Prime Numbers : An introductory page on the theory of prime numbers with proofs of some important theorems including the infinitude of primes, Euclid's algorithm and the fundamental theorem of arithmetic.
		Introductory Prime Number Theory Resources : Notes and links compiled by Mark Watkins on the relation of the Riemann zeta function to the distribution of prime numbers.
	www.webinfosearch.com /Top/Science/Math/Number_Theory/Prime_Numbers (1401 words)

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