
	Where results make sense

Topic: Squarefree

	Theory Mu (Isabelle2004: April 2004)
		0 < d and d dvd n and ~squarefree d} = {}"; by (auto); lemma squarefree_subset1_finite: "(0::int) < n ==> finite {d.
		0 < d and d dvd n and ~ squarefree d}"; by (insert squarefree_subset1_finite [of n] squarefree_subset2_finite [of n], auto); moreover have "{d.
		(0::int) < d and d dvd n and ~squarefree d}"; by (insert prems, auto simp add: squarefree_subset2_finite); ultimately have "setsum mu {d.
	www.contrib.andrew.cmu.edu /~avigad/isabelle/NumberTheory/Mu.html (3169 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The squarefree numbers have a density of 6/pi^2 by a theorem of Gegenbauer.
		(If f(n) is the number of squarefree numbers in 1,...,n, then the limit as n goes to infinity of f(n)/n is 6/pi^2.) Any set of natural numbers having a positive density has arbitrarily long arithmetic sequences in it by a theorem of Szemeredi.
		Let p be a prime that is not a divisor of d; find positive integers x and y such that (p^2)x-dy = 1; let n = ay; then a+nd = (p^2)ax is not squarefree.
	www.math.niu.edu /~rusin/papers/known-math/99/sqfree (248 words)

	StudyWorks! Online : Eight Million Dollar Problems (Site not responding. Last check: 2007-10-14)
		If we factor any of these squarefree integers into primes, no prime will be repeated in the factorization.
		Of the squarefree integers up to 30, exactly 8 are red and 11 are blue.
		The Riemann Hypothesis says roughly that, of the squarefree integers up to an arbitrarily large number N, the discrepancy between the quantities of red numbers and blue numbers is never very large.
	www.studyworksonline.com /cda/content/article/0,,NAV4-44_SAR193,00.shtml (704 words)

	Introduction
		For every squarefree integer d (not 0 or 1) there is a unique quadratic field Q(sqrt(d)); for any integer k the field Q(sqrt(k^2d)) isomorphic to Q(sqrt(d)).
		By default, elements of the quadratic field Q(sqrt(d)) with m squarefree are printed in the form 1/b(x + ysqrtd), where d is a string denoting the integer d and b, x, y are integers; the effect of the above procedure is to replace the string sqrtm by the string s.
		Similarly, for an order O of conductor f in a quadratic field elements are printed by default in the format x + yfepsd, where epsd is the standard integral basis generator; the effect of the procedure is to replace the string f*epsd by the string s.
	www.math.uiuc.edu /Software/magma/text343.html (766 words)

	The CTK Exchange Forums
		A number is defined as squarefree if none of it's prime factors are repeated.
		My contention is that if F is a squarefree number and a and b are relatively prime then the number N = a^2 + F * b^2 is also a squarefree number.
		For example 3^2 + 5 * 7^2 = 254 = 2 * 127 which is squarefree since 2 and 127 are both primes and are not repeated.
	www.cut-the-knot.com /htdocs/dcforum/DCForumID4/605.shtml (1095 words)

	PlanetMath: square-free number
		some questions about squarefree numbers by Wkbj79 on 2003-03-13 02:35:57
		Let s(n) denote the function that takes positive integers as its input and yields the number of (positive) squarefree numbers less than or equal to its input as its output.
		Re: some questions about squarefree numbers by Wkbj79
	planetmath.org /encyclopedia/SquareFree.html (183 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Subject: Re: "Squarefree Polynomials" Date: Fri, 14 Jul 2000 10:51:04 +0200 Newsgroups: sci.math Summary: [missing] Dr.
		I would call a polynomial like 4x^2+4 squarefree, and then almost all poynomials with integer coeficients are squarefree.
		However, if the base ring has zero divisors or is noncommutative, polynomials have much more factors than over fields, and their probability to be squarefree should decrease.
	www.math.niu.edu /~rusin/known-math/00_incoming/sqf (381 words)

	FPSAC01: Abstract for Paper 36 (Site not responding. Last check: 2007-10-14)
		First, we show the existence of squarefree quadratic initial ideals for configurations arising from the set of positive roots of root systems together with the origin.
		Second, we discuss squarefree initial ideals and unimodular coverings for configurations arising from the set of positive roots of root systems together with the origin, without the origin, and thier subconfigurations.
		It is well-known that corresponding affine semigroup ring is normal if the toric ideal has a squarefree initial ideal or if the convex hull of the configuration has a unimodular covering.
	math.la.asu.edu /~fpsac01/PROGRAM/36.html (120 words)

	All Fermat numbers are squarefree (and more)
		Definition: A squarefree number is a number that is not divided by any square.
		Quoting the first document mentioned above, page 8 "At the moment, there are no squarefreeness tests that are essentially faster than factoring." Of course, not all factoring algorithms require a previous squarefreeness test.
		From both lemmata (2.2 and 2.3) it follows that ß(a non squarefree Fermat number) would simultaneously have and have not odd prime factors, which is impossible, following that non squarefree Fermat numbers do not exist.
	www.dybot.com /numbers/sqfree.htm (4373 words)

	Squarefree Values of Multivariable Polynomials - Poonen (ResearchIndex)
		, x n ], we compute the density of x such that f(x) is squarefree, assuming the abc conjecture.
		Bjorn Poonen, Squarefree values of multivariable polynomials, in preparation, 2001.
		Squarefree Values of Multivariable Polynomials - Poonen (2001)
	citeseer.ist.psu.edu /poonen01squarefree.html (495 words)

	hw6ans (Site not responding. Last check: 2007-10-14)
		An integer is called squarefree if it is not divisible by the square of any prime number.
		The first factor is clearly a perfect square, and the second term is a squarefree integer because all of the exponents in the standard factored form are 1.
		By generalizing from the generic particular, any integer greater than 1 can be written as as the product of a squarefree integer and a perfect square.
	www.cs.umd.edu /class/spring2004/cmsc250/hw/hw6ans (556 words)

	On Gaps Between Squarefree Numbers II - Filaseta, Trifonov (ResearchIndex)
		ABC Allows Us to Count Squarefrees - Granville (1998)
		0.2: Squarefree Values Of Polynomials All Of Whose Coefficients..
		2 284--295 (context) - Trifonov, squarefree et al.
	citeseer.ist.psu.edu /filaseta92gaps.html (516 words)

	Conjecture 12. Are Mersenne and Fermat numbers square free ?
		A "squarefree number" is a composite number which has no repeated factor.
		ß(any non squarefree number) has at least one odd prime factor.
		I believe that in general, the postulate in section 4.2 is not true for any Wieferich prime.
	www.primepuzzles.net /conjectures/conj_012.htm (292 words)

	nuwen.net - Random Work
		We'll call a squarefree sequence one which is not square.
		Length two: The sequences 01 and 10 are squarefree; 00 and 11 are not.
		Length three: 010 and 101 are squarefree, but 000, 001, 011, 100, 110, and 111 are not squarefree (as they contain either [0][0] or [1][1]).
	nuwen.net /work.html (4242 words)

	Prime Curios!: 0.6079271018540266286... (Site not responding. Last check: 2007-10-14)
		= the asymptotic density of squarefree numbers (also called quadratfrei), those whose prime decomposition contains no repeated factors.
		The number 1 is by convention taken to be squarefree.
		The squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15,...
	primes.utm.edu /curios/page.php?number_id=1326 (91 words)

	Quadratic fields with cyclic 2-class group
		a) among the complex fields with squarefree radicands 0 > R > -500,
		I found the minimal radicands with the given class number h (and with discriminant d):
		a) a squarefree radicand 0 > R > -2000 with h = 64
	www.algebra.at /AissaDan9.htm (367 words)

	MathLinks Math Forum :: View topic - The idea is very nice
		(this is same as the problem, since if the product of primes is always squarefree, occurs once, and if it exceeds n then [n/product of primes]=0.
		next let f(n) = [n/q1]...+[n/q(k)], g(n) be the number of squarefree divisors of n (not including n and 1).
		but its also obvious that g(n) is even iff n is not squarefree.
	www.mathlinks.ro /Forum/topic-6588.html (463 words)

	Square-free Gaps (Site not responding. Last check: 2007-10-14)
		His idea was to find them by prescribing a repeated prime factors for each term and using the Chinese Remainder Theorem (see also this web page) to obtain a number.
		More precisely, he prescribed all but five of the moduli and then tested the last moduli, up to the first 1000 primes, to check if the number is squarefree.
		A simple method to show that N is not squarefree is to find a prime factor of N whose square divides N.
	www.marmet.ca /louis/sqfgap (2394 words)

	Matches for: Author=filaseta
		Konyagin, Sergei Squarefree values of polynomials all of whose coefficients are $0$ and $1$.
		Filaseta, Michael On the distribution of gaps between squarefree numbers.
		Filaseta, Michael Short interval results for squarefree numbers.
	www.math.sc.edu /~filaseta/public.html (490 words)

	On the Distribution of Gaps between Squarefree Numbers - Filaseta (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		On the Distribution of Gaps between Squarefree Numbers (1993)
		Filaseta, M. On the distribution of gaps between squarefree numbers.
		@misc{ filaseta93distribution, author = "M. Filaseta", title = "the distribution of gaps between squarefree numbers", text = "Filaseta, M. On the distribution of gaps between squarefree numbers.
	citeseer.ist.psu.edu /filaseta93distribution.html (452 words)

	Creation of Structures (Site not responding. Last check: 2007-10-14)
		Given an integer m that is not a square, create the field Q(Sqrt(d)), where d is the squarefree part of m.
		Creation of the order Z[Sqrt(d)] in the quadratic field F=Q(Sqrt(d)), with d squarefree.
		Given a quadratic field F=Q(Sqrt(d)), with d squarefree, create its maximal order.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text742.htm (337 words)

	W.Lang: BinQuadForm.html (web-server October 2003)
		If there is no fundamental solution the output will be [].
		Similar for improper solutions which can only occur if k is not squarefree.
		See A0051170 for positive squarefree numbers (1 is squarefree).
	www-itp.physik.uni-karlsruhe.de /~wl/emptyBinQuadForm.html (529 words)

	Small Groups
		This library has been prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien.
		The groups have been been constructed by the authors of the library with the help of Mike Newman (order p^4), Boris Girnat (order p^5), Mike Newman and Mike Vaughan-Lee (p^6) and Heiko Dietrich (squarefree and cubefree orders).
		The groups of squarefree order and those of an order which factorises in at most 3 primes have been known for a long time.
	www.tu-bs.de /~hubesche/small.html (703 words)

	blog of francois: Bookmarklets and other useful things for IE’s Links bar
		One of the best selections (for the web developer) is to be found at Jesse Ruderman’s Squarefree.
		Most bookmarklets will also work on Mozilla-based browsers (indeed, many are exclusive to it), but many are conveniently bundled into custom toolbars or extensions.
		Unfortunately this does not work on sites where the background colour is set by the stylesheet.
	www.fjordaan.uklinux.net /moveabletype/fblog/archives/000059.html (1711 words)

	Index of /users/tornaria/cnt/twist/current (Site not responding. Last check: 2007-10-14)
		Added imaginary twists of prime level less than 20000.
		Added imaginary twists of squarefree level less than 5000.
		The sample program "twist.gp" has been extended to deal with the composite (squarefree) case.
	www.ma.utexas.edu /users/tornaria/cnt/twist/current (77 words)

	Ed Pegg's Math Games - The Möbius Function (and squarefree numbers)
		In 1831, August Ferdinand Möbius put numbers into 3 bins, as a new type of function.
		Because of the relation to Gaussian integers, I'll vote for G.
		In the image below, the fl pixels represent squarefree Gaussian integers.
	www.maa.org /editorial/mathgames/mathgames_11_03_03.html (1425 words)

	Some UBASIC Programs for use in a Number Theory course
		The Mobius function: =1 if n=1, =(-1)s if n is squarefree
		and has s prime factors, =0 if n is not squarefree
		Gives the remainder when a raised to b is divided by n
	www.unf.edu /~ramm/numbtheory.html (302 words)

	Technorati Tag: squarefree (Site not responding. Last check: 2007-10-14)
		A tag is like a subject or category.
		This page shows blog posts, photos, and links that have been tagged squarefree.
		To contribute to this page, just post to your blog and include this code.
	www.technorati.com /tag/squarefree (206 words)

	The data type real ( real )
		real_roots(const Polynomialand P, listand roots, algorithm_type algorithm = isolating_algorithm, bool is_squarefree = true)
		real_roots(const int_Polynomialand iP, listand roots, algorithm_type algorithm = isolating_algorithm, bool is_squarefree = true)
		diamond(rational l, rational u, const int_Polynomialand iP, algorithm_type algorithm = isolating_algorithm, bool is_squarefree = true)
	www.algorithmic-solutions.info /leda_manual/real.html (1333 words)

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