Square-free integer

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Topic: Square-free integer

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	Square number - Wikipedia, the free encyclopedia
		In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer.
		This average is the variance, and its square root is the standard deviation -- in finance, the volatility.
		It follows, in particular, that no square root can be taken of a negative number within the system of real numbers.
	en.wikipedia.org /wiki/Square_number (535 words)

	square_free_part_0
		This routine find the square-free part of the integer, which is given as a prime factors list la -- a MONOPOLY containing the prime factorization of the integer.
		lb and lc return MONOPOLYs containing the prime factorization of the square-free part and square-root of the square part, respectively.
		That is, lb has no prime (including -1), occurring with an exponent > 1.
	www.mathe2.uni-bayreuth.de /axel/help/square_free_part_0_help.html (78 words)

	Square root - Wikipedia, the free encyclopedia
		Square roots of positive integers are often irrational numbers, i.e., numbers not expressible as a ratio of two integers.
		In geometrical terms, the square root function maps the area of a square to its side length.
		Thus defined, the square root function is holomorphic everywhere except on the non-positive real numbers (where it isn't even continuous).
	en.wikipedia.org /wiki/Square_root (1274 words)

	Integer square root - Wikipedia, the free encyclopedia
		In number theory (a branch of mathematics), the integer square root (isqrt) of a positive integer n is the positive integer m which is the greatest integer less than or equal to the square root of n,
	en.wikipedia.org /wiki/Integer_square_root (224 words)

	Square-free
		In mathematics, an integer n is called square-free iff no perfect square except 1 divides n.
		Equivalently, n is square-free iff in the prime factorization of n, no prime number occurs more than once.
		If Q(x) denotes the number of square-free numbers less than or equal to x, then
	www.fastload.org /sq/Square-free.html (238 words)

	Square-free Encyclopedia Article, Definition, History, Biography
		In mathematics, a square-free integer is one divisible by no perfect square, except 1.
		The above information is licensed under the GNU Free Documentation License.
		List Your Site Here for Free*: Simply link to this page from your website and a reciprocal link back to your referring page will automatically appear above.
	www.artquilt.com /encyclopedia/Square-free (512 words)

	Square Free
		In mathematics, a square-free integer n is one divisible by no perfect square, except 1.
		Toll Brothers at Naval Square will also provide tours of its newest community, Naval Square, and they will have limited free parking.
		All text is available under the terms of the GNU Free Documentation License [see Copyrights for details].
	www.wikiverse.org /square-free (364 words)

	Square number - Wikipedia, the free encyclopedia
		In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer.
		If rational numbers are included, then the ratio of two square integers is also a square (e.g.
		The sum of two consecutive square numbers is a centered square number.
	en.wikipedia.org /wiki/Square_number (672 words)

	Square (algebra) - Wikipedia, the free encyclopedia
		A positive integer that is the square of some other integer, for example 25 which is 5
		This average is the variance, and its square root is the standard deviation -- in finance, the volatility.
		In algebra, the square of a number is that number multiplied by itself.
	en.wikipedia.org /wiki/Square_(algebra) (256 words)

	Square wave - Wikipedia, the free encyclopedia
		In contrast to the sawtooth wave, which contains all integer harmonics, the square wave contains only odd integer harmonics.
		Square waves are universally encountered in digital switching circuits and are naturally generated by binary (two-level) logic devices.
		The ratio of the high period to the total period of a square wave is called the duty cycle.
	en.wikipedia.org /wiki/Square_wave (727 words)

	Square (algebra) - Wikipedia, the free encyclopedia
		A positive integer that is the square of some other integer, for example 25 which is 5
		In algebra, the square of a number is that number multiplied by itself.
		is equal to the area of a square of edge length x.
	www.wikipedia.org /wiki/Square_%28algebra%29 (106 words)

	Perfect square - Wikipedia, the free encyclopedia
		a positive integer which is the square of some other integer, i.e.
		In general, the product of two numbers is equal to the square of their average minus their difference from the average squared.
		This is not the same as a magic square.
	en.wikipedia.org /wiki/Perfect_square (205 words)

	Xinyu Sun's Result on Ternary Square-Free Words
		Uwe Grimm, Improved Bounds on The Number of Ternary Square-Free Words, J. Integer Seq.
		The article presents a new lower bound on the number of n-letter ternary square-free words: 110
		Binary words are the words from two-letter alphabet {0, 1}, whereas ternary words are from three-letter alphabet {0, 1, 2}.
	www.math.tamu.edu /~xsun/ternarysf/ternary_square_free.htm (231 words)

	Square number - Wikipedia, the free encyclopedia
		In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer.
		The sum of two consecutive square numbers is a centered square number.
		A square number is also the sum of two consecutive triangular numbers.
	en.wikipedia.org /wiki/Square_number (535 words)

	Solving General Pell Equations
		Methods for factoring integers include Pollard's number field sieve, the special number field sieve, Pomerance's quadratic sieve, the Brillhart-Morrison continued fraction factoring algorithm, D. Shanks' square-free factorization, Pollard's rho method, and the elliptic curve method.
		= N, for d a positive integer, not a square, and N a nonzero integer.
		This algorithm is at the heart of all of the algorithms to solve Pell equations presented here.
	members.aol.com /jpr2718/pellold.html (535 words)

	Squaring the square - Wikipedia, the free encyclopedia
		A square with sides equal to a unit length multiplied by an integer is called an integral square.
		They transformed the square tiling into an equivalent electrical circuit, by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit.
		A "perfect" squared square is such a square such that each of the smaller squares has a different size.
	en.wikipedia.org /wiki/Squaring_the_square (437 words)

	Missing square puzzle - Wikipedia, the free encyclopedia
		The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers.
		The key to the puzzle is the fact that neither of the 13x5 "triangles" has the same area as its component parts.
		Many other geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.
	en.wikipedia.org /wiki/Missing_square_puzzle (274 words)

	Creation of Quaternion Algebras
		The rational quaternion algebra with square-free discriminant N. The integer N must be a product of an odd number of primes, i.e.
		In the case of this constructor the algebra generators are of trace zero and are pairwise anticommuting.
		The rational quaternion algebra Q< i, j >, where Z[i] and Z[j] are quadratic suborders of discriminant D_1 and D_2, respectively, and Z[ij - ji] is a quadratic suborder of discriminant D_3 = D_1 D_2 - T^2.
	www.math.lsu.edu /magma/text865.htm (683 words)

	Squaring the square - Wikipedia, the free encyclopedia
		A square with sides equal to a unit length multiplied by an integer is called an integral square.
		They transformed the square tiling into an equivalent electrical circuit, by considering the squares as resistors that connected to their neighbors at their top and bottom edges, and then applied Kirchhoff's circuit laws and circuit decomposition techniques to that circuit.
		A "simple" squared square is one where no subset of the squares forms a rectangle or square, otherwise it is "compound".
	en.wikipedia.org /wiki/Squaring_the_square (462 words)

	Polynomial factorization
		Berlekamp-Hensel algorithm for factorization of "square free" polynomials in one variable with integer coefficients
		consider polynomials with integer coefficients (it is simple to convert a polynomial with rational coefficients into a problem in this domain)
		Next: Quantifier elimination Up: Algorithms for algebraic computation Previous: Ordinary differential equations
	www-troja.fjfi.cvut.cz /~liska/ca/node47.html (462 words)

	PlanetMath: radical of an integer
		The radical of a square-free number is itself.
		This is version 6 of radical of an integer, born on 2001-10-15, modified 2004-02-26.
		"radical of an integer" is owned by KimJ.
	planetmath.org /encyclopedia/SquareFreePart.html (462 words)

	On-Line Encyclopedia of Integer Sequences - Wikipedia, the free encyclopedia
		Sequence A073502, the magic constant for n X n magic square with prime entries (regarding 1 as a prime) with smallest row sums, is an example of a sequence with offset 3, and A072171, "Number of stars of visual magnitude n." is an example of a sequence with offset -1.
		The 1973 Handbook of Integer Sequences contained about 2400 sequences, which were numbered by lexicographic order (the letter M plus 4 digits, zero-padded where necessary), and the 1995 Encyclopedia of Integer Sequences contained 5487 sequences, also numbered by lexicographic order (the letter N plus 4 digits, zero-padded where necessary).
		Every sequence in the OEIS has a serial number, a six-digit positive integer, prefixed by A (and zero-padded on the left prior to November 2004).
	en.wikipedia.org /wiki/On-Line_Encyclopedia_of_Integer_Sequences (4149 words)

	Problem 28.- K least consecutive numbers n such that are not squarefree
		This kind of sequences may also be named as "gaps of square free consecutive numbers" and also as "sequences of consecutive numbers whose Möbius function is zero".
		This table corresponds to the sequence A051681 of the Sloane's On-Line Encyclopedia of Integer Sequences.
		Here is the list of the sequences found up to November 1999 for K=1 to 15.
	www.primepuzzles.net /problems/prob_028.htm (456 words)

	PlanetMath:
		square-free part (radical of an integer) owned by KimJ
		smaller integer greater or equal than (ceiling) owned by drini
		sets that do not have an infimum owned by sleske
	bshanksserver.dyndns.org /bshanks/symSysHonorsThesis/talkSlides/planetmathIndex.html (456 words)

	Xinyu Sun's Result on Ternary Square-Free Words
		Uwe Grimm, Improved Bounds on The Number of Ternary Square-Free Words, J. Integer Seq.
		However, there are infinitely many ternary square-free words.
		A word is square-free if it does not contain two identical consecutive subwords, i.e.
	www.math.tamu.edu /~xsun/ternarysf/ternary_square_free.htm (456 words)

	W. F. Hammond: Presentations
		If B + S-D is true, then there is an algorithm for deciding whether or not a given square-free integer d is a congruent number.
		Congruent numbers: square-free representatives for the areas of similarity classes of rational right triangles.
		Of these 1906 give rise to the first occurrence of a (square-free) congruent number via the rational parameterization.
	math.albany.edu:8000 /math/pers/hammond/Presen/rsumo.html (1039 words)

	Talk:Quadratic field - Information
		For example, is $\mathbb{Q}(\sqrt{2/3})$ a quadratic field, and is it expressible as $\mathbb{Q}(\sqrt{d})$ for d a square-free integer?
		That is, it is being claimed that rationals up to squares of rationals are represented uniquely by square free integers.
		I think it's a pretty short exercise to show that all quadratic extension of Q are of the form Q(sqrt(d)), for square-free d, so maybe this proof could just be spelled out (I think it's only a couple lines).
	www.book-spot.co.uk /index.php/Talk:Quadratic_field (288 words)

	Polynomial factorization
		Berlekamp-Hensel algorithm for factorization of "square free" polynomials in one variable with integer coefficients
		Berlekamp algorithm for factorization of "square free" polynomials in one variable modulo a prime
		is "square free", i.e., it does not have a factor which is the second or higher power of a polynomial, if and only if
	www-troja.fjfi.cvut.cz /~liska/ca/node47.html (119 words)

	Unimodular matrix - Wikipedia, the free encyclopedia
		A totally unimodular matrix is an integer matrix for which every square non-singular submatrix is also unimodular.
		In mathematics, a unimodular matrix is a square matrix with determinant +1 or -1.
		An integer program whose constraint matrix is totally unimodular and whose right hand side is integral, can be solved efficiently since its LP relaxation gives rise to integer solutions.
	en.wikipedia.org /wiki/Unimodular_matrix (119 words)

	Quadratic sieve - Wikipedia, the free encyclopedia
		The algorithm attempts to set up a congruence of squares modulo n (the integer to be factorized), which often leads to a factorization of n.
		The naïve approach to finding a congruence of squares is to pick a random number, square it, and hope the least non-negative remainder modulo n is a perfect square (in the integers).
		The quadratic sieve algorithm (QS) is a modern integer factorization algorithm and, in practice, the second fastest method known.
	en.wikipedia.org /wiki/Quadratic_sieve (2116 words)

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Topic: Square-free integer

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