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Topic: Gaussian primes

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	NationMaster - Encyclopedia: Gaussian integer (Site not responding. Last check: 2007-11-01)
		The norm of a Gaussian integer is the natural number defined as In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector.
		Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not.
		The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.
	www.nationmaster.com /encyclopedia/Gaussian-integer (1377 words)

	Complex primes
		Gaussian Integer's behave in many respects like the "rational" integers (the term often used to distinguish the "normal" integers from the Gaussian Integer's); indeed, the set of Rational Integers is a subset of the set of Gaussian Integer's.
		If Z is a Gaussian Integer and u is a unit, then the Gaussian Integer uZ is called an associate of Z. A Gaussian* Integer is said to be prime if it is not 0 nor a unit, and if its only divisors over the Gaussian Integer's are the units and its associates.
		They make no reference to Gaussian integers or primes, but their definition for a complex prime number coincides with the definition of a Gaussian prime, except they exclude the Gaussian primes corresponding to the "p = 4n + 3" primes*.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH036.HTM (720 words)

	What are Gaussian Primes? -- from Harry J. Smith
		A Gaussian prime is a Gaussian integer p with exactly 8 divisors: p, −p, pi, −pi, 1, −1, i, and −i.
		The prime integer 5 is not a Gaussian prime because 5 can be written as (1 + 2i) * (1 − 2i) = (2 + i) * (2 −i).
		The number 5 has 8 Gaussian prime integer divisors 1 + 2i, 1 − 2i, −1 + 2i, −1 − 2i, 2 + i, 2 − i, −2 + i, and −2 − i.
	www.geocities.com /hjsmithh/GPrimes/GPriWhat.html (519 words)

	Gaussian integer - Wikipedia, the free encyclopedia
		Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i).
		Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not.
		The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.
	en.wikipedia.org /wiki/Gaussian_integer (331 words)

	Prime number - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Prime_number (2826 words)

	Gaussian integer - Definition, explanation
		Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i).
		If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm.
		Gaussian Integers, Fermat's Last Theorem Blog traces the history of Fermat's Last Theorem from Diophantus of Alexandria to Andrew Wiles.
	www.calsky.com /lexikon/en/txt/g/ga/gaussian_integer.php (379 words)

	Prime number - ExampleProblems.com
		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.exampleproblems.com /wiki/index.php/Prime_number (3311 words)

	Gauss Product (Site not responding. Last check: 2007-11-01)
		A Gaussian integer is a complex number with integer coefficients.
		Gaussian integers are 1 of only 10 number systems that have unique factorization into primes, these being the normal integers, and the 9 Heegner number systems, -1 (Gaussian), -2, -3 (Eisenstein), -7, -11, -19, -43, -67 and -163.
		Here, p is over the Gaussian primes, and x* and y* are over the integers, the *'s indicate that x=y=0 is not allowed.
	www.users.globalnet.co.uk /~perry/maths/gaussproduct/gaussproduct.htm (291 words)

	Math Games: Prime Megagap
		An example of a prime gap of size 20 is found between primes 887 and 907.
		With numbers of this size, the average gap is ln(887) ~ 6.78, by the prime number theorem.
		Probable primes are prime with a 99.99999999999999% certainty, and this battery of tests never fails for numbers under 10^16.
	www.maa.org /editorial/mathgames/mathgames_01_25_04.html (851 words)

	ACM Sigplan Notices 28, 11 (Nov 1993), 22-27.
		Once the basic concepts are in hand, we should revel in the polymorphism of Gaussian integers as pairs of integers, as well as atomic elements of the Gaussian ring, much as we revel in the dual role of C "ints" as both integers and bit strings.
		We know that for a Gaussian integer divisor m+ni, we need m^2+n^2 distinct remainders, so the most elegant choice of representative remainders is a square of area m^2+n^2 which is tilted at an angle of atan(n/m);[5] equivalently, we consider the remainder fraction r/d to reside in an upright square of area 1.
		Gaussian primes of the form m+i appear to be moderately common--e.g., there are about 50 whose norms are less than 100,000.
	home.pipeline.com /~hbaker1/Gaussian.html (6038 words)

	American Mathematical Monthly, The: A stroll through the Gaussian primes (Site not responding. Last check: 2007-11-01)
		This problem is sometimes called the Gaussian moat problem, since one way of establishing a walk's nonexistence is to present a sufficiently wide moat (region of composites) that completely surrounds the origin.
		The ring of Gaussian integers, denoted Z[i], consists of integers in the field Q(i); they have the form a + bi where a, b epsilon Z and i = (square root of) -1.
		There are Gaussian integers m-=0 and b such that the real and imaginary parts of m are relatively prime and the Gaussian integer z is on this line if and only if there is an x epsilon(Zeta) such that z = mx + b.
	www.findarticles.com /p/articles/mi_qa3742/is_199804/ai_n8786812 (1608 words)

	Pierre-Simon Laplace - Metaweb (Site not responding. Last check: 2007-11-01)
		He was the first to publish the value of the Gaussian integral: a complex number whose real and imaginary part are both integers.
		Some prime numbers are not Gaussian primes; for example 2=(1+i)(1-i) and 5=(2+i))(2-i).
		Those prime numbers which are congruent to 3 mod 4 are Gaussian primes; those which are congruent to 1 mod 4 are not.
	www.metaweb.com /wiki/wiki.phtml?title=Pierre-Simon_Laplace (888 words)

	AMERICAN MATHEMATICAL MONTHLY - APRIL 1998
		The Gaussian question is much more complex because of the additional dimension.
		This problem is sometimes called the Gaussian moat problem, since one way of establishing a walk's nonexistence is to present a sufficiently wide moat (region of composites) that completely surrounds the origin.
		We also show that there exists a real Gaussian prime that is isolated in a neighborhood of radius k for any positive k.
	www.maa.org /pubs/monthly_apr98_toc.html (639 words)

	Z[i]
		The norm N(α) of a Gaussian integer α = a + bi is defined as the integer a
		Every non-zero Gaussian integer which is not a unit can be written as a product of prime elements.
		CommentThe Gaussian integer α = a + bi and its conjugate σ(α) = a - bi are the roots of the quadratic equation
	wwwmaths.anu.edu.au /DoM/thirdyear/MATH3301/gaussian.html (879 words)

	Luboš Motl's reference frame: New record prime
		Note that if you find the first Mersenne prime with at least 10 million digits, you will win at least 1/2 of the $100k award from the EFF foundation.
		Mersenne primes are in Topological Geometrodynamics framework the most interesting primes since they correspond to most important p-adic length scales.
		Also Gaussian primes associated with complex integers are important in TGD framework.
	motls.blogspot.com /2005/02/new-record-prime.html (1098 words)

	Gaussian Integers
		This forms the complex integers, or gaussian integers, named after the mathematician Carl Gauss (biography), who dabbled in just about everything.
		Thus 4+i and 4-i are both prime in the gaussian integers.
		There are infinitely many gaussian primes, via the same argument as before.
	www.mathreference.com /num,gi.html (845 words)

	The Top Twenty: Gaussian Mersenne norm
		Recall that the Mersenne primes are the primes of the form 2
		is prime or b=0 and a is a prime congruent to 3 (mod 4).
		For example, the prime factors of two are 1+i and 1-i, both of which have norm 2.
	primes.utm.edu /top20/page.php?id=41 (554 words)

	[No title]
		1 + i is a prime, and so are its associates —1+i, 1 — i, and —1 — i.
		Any rational prime p that satisfies p (3 (mod 4) is a Gaussian prime, and so are its associates —p, pi, and —pi.
		Primes of this form come in eights: There are a + bi and its associates, and the complex conjugate a — bi and its associates.
	www.swarthmore.edu /NatSci/wstromq1/numbers/GaussianPrimes.doc (301 words)

	Number Theory
		As we have seen, rational primes may or may not be Gaussian primes.
		Primes of this form come in eights: There are a + bi and its associates, and the complex conjugate a – bi and its associates.
		We don’t have ordering in the Gaussian integers, but we have the norm, which turns out to be just as good.
	www.swarthmore.edu /NatSci/wstromq1/numbers/GaussianPrimes.htm (858 words)

	Neglected Gaussians
		It's always posssible to factors a Gaussian Integer into positive Gaussian Primes of the form x+yi with x>0 and y>=0, and one of the four roots of unity.
		There seems to be an infinite number of primes of this form for any n.
		For (8+i)^n + 2, the expression is prime for n=1,2,3,4,5.
	www.mathpuzzle.com /Gaussians.html (723 words)

	Mersenne: A new series of Mersenne-like Gaussian primes
		As with the Mersennes, n must be prime else s[n] is composite.
		For n = 2, s[n] = (1+i)^2 - 1 = 2i - 1, with modulus (2i-1)(-2i-1) = 5, which is prime.
		It is easy to prove that any prime factor of A[n] is of the form 4kn+1.
	www.mail-archive.com /mersenne@base.com/msg05162.html (343 words)

	Gaussian Prime -- from Wolfram MathWorld
		The above plot of the complex plane shows the Gaussian primes as filled squares.
		The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43,...
		The cover of Bressoud and Wagon (2000) shows an illustration of the distribution of Gaussian primes in the complex plane.
	mathworld.wolfram.com /GaussianPrime.html (221 words)

	phorum - Foro de Álgebra - Primos en C (complejos) (Site not responding. Last check: 2007-11-01)
		Gaussian Integer and u is a unit, then the Gaussian Integer u*Z is called an
		Gaussian prime, except they exclude the Gaussian primes corresponding to the
		primes of the form 4n + 3 in the regular integers cannot be factored in the
	www.us.es /foros/read.php?f=7&i=73&t=73 (792 words)

	Number theory
		By definition, a twin prime is a prime number n such that n+2 is also a prime number.
		The first member of each pair is the factor (a Gaussian integer) and the second member denotes the power to which this factor should be raised.
		A prime element x of a ring is divisible only by the units of the ring and by associates of x.
	yacas.sourceforge.net /refchapter4.html (1987 words)

	Arithmetic, Numeration, Number Theory - Numericana
		The so-called Gaussian integers are complex numbers of the form a+ib, where a and b are integers.
		Ordinary prime integers are not necessarily prime among Gaussian integers.
		Actually, a prime integer is a Gaussian prime if and only if it's congruent to 3 modulo 4.
	home.att.net /~numericana/answer/numbers.htm (7666 words)

	wikien.info: Main_Page (Site not responding. Last check: 2007-11-01)
		(This is sequence [A000040] in OEIS; see list of prime numbers for the first 500 primes.) The set of all prime numbers is sometimes denoted by ℙ, a flboard bold P.
		− 1 are known as Mersenne primes, while primes of the form [2^{2^n} + 1] are known as Fermat primes.
		As another example, we can extend the integers to the Gaussian integers Z[i], that is, complex numbers of the form a + bi with a and b in Z.
	www.alanaditescili.net /index.php?title=Prime_number (2563 words)

	Prime Percolation - Vardi (ResearchIndex) (Site not responding. Last check: 2007-11-01)
		Percolation theory predicts that for a low enough density of random Gaussian integers no walk exists, which suggests that no such walk exists along prime numbers, since they have arbitrarily small density over large enough regions.
		Vardi, "Prime Percolation," Experimental Mathematics 7:3 (1998), 275--288.
		2 Irregularities in the distribution of primes in short interv..
	citeseer.ist.psu.edu /vardi98prime.html (640 words)

	Computations for Katz' Observations (Site not responding. Last check: 2007-11-01)
		This document presents some computations on the "cumulative distribution" function of normalized spacings between angles of Gaussian primes, as defined by N. Katz in a lecture given at Princeton on October 12, 1994.
		This outputs a vector of all the angles of gaussian primes of norm between
		The thick curve is the cumulative distribution function of the spacings for the gaussian primes, while the thin curve is the possible limiting exponential curve.
	www.math.okstate.edu /~wrightd/numthry/katzcomp.html (254 words)

	Gaussian primes (Site not responding. Last check: 2007-11-01)
		The Gaussian integers are complex numbers of the form a+bi where a and b are (standard) integer numbers.
		These numbers can be factored in Gaussian primes.
		Move the center by typing a new complex number in both input boxes (up to 7 digits each) and press the return key.
	www.alpertron.com.ar /GAUSSPR.HTM (112 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)

	HAKMEM -- NUMBER THEORY, PRIMES, PROBABILITY -- DRAFT, NOT YET PROOFED
		The number of primes in 10^12 + 1 to 10^12 + 10018 is 335; the prime number theorem predicts 363 in this range.
		Primes with 4 sub-cycles seem to always be of form 4 K + 1, and seem to have lengths
		Attention was directed to primes which are 1 or 9 mod 40 but have 1 or 4 subcycles.
	www.inwap.com /pdp10/hbaker/hakmem/number.html (2842 words)

	Essays/UniqueFactors - J Wiki
		Basically, pcp considers a number to be prime only if there are no more than three complex integers (one, itself and 0j1) which can be multiplied together to produce the number.
		Regular prime numbers (where the domain is solely positive integers) are called "Rational Primes" for contrast.
		Eisenstein primes A concept somewhat analogous to Gaussian Primes, but based on the third root of 1 (which might prevent factorizing negative integers, except that negative 1 is given special treatment such that the negative of a prime number is also considered prime) rather than on the square root of _1.
	www.jsoftware.com /jwiki/Essays/UniqueFactors (800 words)

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