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Topic: Continued fraction factorization

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Integer factorization

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	Factorization (Site not responding. Last check: 2007-10-11)
		a rectangle with the dimensions of the factorization.
		Factorization `QR` FactorizationThe traditional algorithm for QR factorization is based on the use of elementary Householder matrices of the general form.
		Factorization QR FactorizationGiven an matrix, we seek the factorization, where is an orthogonal matrix, and is an upper triangular matrix.
	www.factoring.webfinance.ws /factorization/index.php (1935 words)

	Continued Jt \| Continued Proportion (Site not responding. Last check: 2007-10-11)
		The ever developing web means continued jt business people reach new goals with the increased business A major advantage that online continued from 10 retailers will have over the mall and physical store type factoring continued fractions enterprises is the reduced cost of conducting business operations..
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		Our nursing online continued education website is relatively new so we have not yet managed to write lots of information, but what we have done so far is researched the very best name continued sites on the net.
	dngu.info /continued-jt.htm (356 words)

	The Mean When a is Irrational
		One way to study such approximations is with the theory of continued fractions, and the proof of Part 2 of Theorem 1 is based on results that can be found in the book Continued Fractions by Khinchin [3].
		Luckily, the theory of continued fractions allowed us to obtain an upper bound on the contribution of these eigenvalues, at least in some cases.
		Continued fractions provide a method to find the best rational approximations, or convergents, of an irrational number, and this procedure yields many results about the general problem of approximating irrational numbers.
	www.math.washington.edu /~ndbs/research/UAevals/node5.html (690 words)

	Continued fraction factorization (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-11)
		In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm.
		The continued fraction method is based on Dixon's factorization method.
		Since this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious).
	www.danceage.com.cob-web.org:8888 /biography/sdmc_Continued_fraction_factorization (104 words)

	Current Research, McMath, Trident
		Also observe that if we continue the expansion after the factorization is obtained, the sequences repeat, except in the opposite order and paired differently (Figure 4).
		Based on this symmetry, any fast test to determine whether or not the continued fraction expansion is in the correct direction, that is, whether or not it has not passed the factorization yet, would provide a faster factorization algorithm by performing a binary search (Figure 5).
		However, polynomial time factorization could be possible even without this frequency of response, since all that is required for polynomial time factorization is that the time required for this test of direction be a polynomial function of the number of digits.
	web.usna.navy.mil /~wdj/mcmath/node3.html (673 words)

	In today’s world of e-commerce and mass digital communication, there is a clear need for the ability to securely ... (Site not responding. Last check: 2007-10-11)
		As the remainder is a proper fraction with a as the denominator, r
		In other similar cases, the factors of the numbers might be assumed to be closer to the middle of the set of possible divisors as to not make the factoring too easy.
		While the algorithm is longer than that for brute force factoring, the continued fraction factoring method has shown itself in examples to be a more efficient process for determining factors of a large number.
	www.mit.edu /~rkabir/research/Fractions.htm (3138 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-11)
		The factors in the denominator will cancel with some in the numerator, but what is left will still be a useful factorization of N into two parts.
		Continue this until you find a d(n) with 1 < d(n) < N. Then d(n) is a factor of N, which is then split into two smaller factors d(n) and N/d(n).
		The worst-case time to factor a composite N is about sqrt(N) steps, but the average time for a random N is much smaller.
	mathforum.org /library/drmath/view/65801.html (975 words)

	Factorization
		If n is a positive integer whose prime factorization is given by the factorization sequence, return a sequence containing all of the divisors of n, including 1 and n.
		The function will always return the complete prime factorization (in the form of a factorization sequence) of the number n (but it may take very long before it completes); it should be pointed out, however, that the primes appearing in the factorization are only probable primes and a rigorous primality prover has not been applied.
		If the factorization could not be completed, a third sequence is returned, containing composite factors that could not be decomposed with the given values of the parameters.
	www.math.uiuc.edu /Software/magma/text296.html (1091 words)

	Wolfram Research, Inc.
		This package supports two alternative representations: the continued fraction expansion of a real number and the arbitrary base expansion of a rational number in terms of preperiodic and periodic parts.
		Continued fractions also find application in the factorization of integers (see, for example, Chapter 10 in [Rosen]).
		This gives partial quotients in the continued fraction expansion of the square root of 7.
	documents.wolfram.com /v3/AddOns/NumT_ContinuedFractions-.html (311 words)

	Prime Factorizations - Factoring Large Integers into Primes - Numericana
		This is the right idea but it won't do theoretically "as is", because the "least special" prime factors of numbers of bounded size may be used to establish a generous overall upper bound on the running time of any dubiously defined "special-purpose" algorithm (provided it's "general" enough to discover any factor, albeit slowly).
		Long before the notion became relevant to the factorization algorithms discussed here, number theorists had defined a primitive factor of a term in a sequence as a prime divisor of that term which does not divide earlier terms in the sequence.
		This method is intended for the factorization of a fairly large number n, preferably once smaller factors have been removed, using preliminary methods which are more efficient at weeding out small or medium-sized divisors...
	home.att.net /~numericana/answer/factoring.htm (2746 words)

	Continued Fraction -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-11)
		Continued fractions provide, in some sense, a series of "best" estimates for an irrational number.
		Continued fractions can also be used to calculate gear ratios, and were used for this purpose by the ancient Greeks (Guy 1990).
		Continued fractions can be used to express the positive roots of any polynomial equation.
	mathworld.wolfram.com.cob-web.org:8888 /ContinuedFraction.html (1394 words)

	Background, McMath, Trident
		This research will analyze an algorithm for integer factorization based on the use of continued fractions and quadratic forms, primarily intending to produce a runtime analysis of the algorithm but also proving several valuable results about continued fractions.
		Although fast factorization would be a threat to this system, the advance in number theory produced by fast factorization would likely provide a number of alternative secure systems.
		If there are classes of numbers that a fast factorization algorithm does not work on, this would allow designers of the algorithm to increase their security by relying more on these numbers.
	web.usna.navy.mil /~wdj/mcmath/node2.html (1388 words)

	Integer factorization (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-11)
		The factorization is always unique, according to the fundamental theorem of arithmetic.
		If a fast method were found for solving the Integer factorization problem, then several important cryptographic systems would be broken, including the RSA public-key algorithm and the Blum Blum Shub pseudo-random number generator.
		As of 2005, the largest number factored using general-purpose methods as part of public research is RSA-200, which is 663 bits long.
	integer-factorization.iqnaut.net.cob-web.org:8888 (837 words)

	[No title]
		Recent announcements of work in progress indicate that integers of around 116 digits may be factored shortly, using the same network used in the 106-digit case and another variant of the quadratic sieve.
		No algorithms are known to generate class numbers with large prime divisors, although interestingly RSA comes close: class numbers h(-n) of discriminants n with one or two prime factors have a higher probability of having large prime factors than the class numbers of discriminants with only small prime factors.
		Some other factoring methods are applicable to numbers which do not have the form required for an RSA modulus.
	www.eff.org /Privacy/Crypto/Crypto_misc/nist_pubkey_crypt.paper (11164 words)

	Factorization
		If the function succeeds in finding a proper factor, this is returned as the value of q while r returns the quotient n/q.
		Finally, by putting Continue := true one may indicate that if the factorization has not been completed yet, a final attempt using the elliptic curve method with "optimal" parameters should be made (this may be very time consuming).
		By default, after the initial factorization attempts (trial division, Pollard) have not succeeded entirely, ECM is invoked with small bound and few curves (10), then the quadratic sieve.
	www.math.ufl.edu /help/magma/text317.html (1399 words)

	Factorization (Site not responding. Last check: 2007-10-11)
		factorization a Factorization Based Algorithm for Multi-Image Projective...We propose a method for the recovery of projective shape and motion from multiple images of a scene by the factorization of a matrix containing the images...
		At the -th step of the computation, we partition this factorization to the submatrix of asfactorizationmathwords: Linear FactorizationA factored form of a polynomial in which each factor is a linear polynomial.
		To prime factor a number, begin dividing by the smallest possible prime and continue until the quotient is a prime number.
	www.factoring.webfinance.ws /factoring_rules/factorization.php (1983 words)

	Congruence of squares - Wikipedia, the free encyclopedia
		In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.
		Congruences of squares are extremely useful in integer factorization algorithms.
		This congruence is extensively used in, for example, the quadratic sieve, general number field sieve, continued fraction factorization, Dixon's factorization, and so on.
	en.wikipedia.org /wiki/Congruence_of_squares (154 words)

	Sasa Radomirovic - Math 356 - Number Theory - Summer 2004
		That formula also proved useful for proving several properties of the convergents which culminated in the proof of the fact that the odd convergents are monotonely decreasing while the even convergents are monotonely increasing, and every odd convergent is bigger than any even convergent.
		We then proved that all infinite simple continued fractions are irrational, and described a procedure to produce infinite simple continued fractions for irrational numbers.
		We mentioned, but didn't prove the uniqueness of infinite simple continued fractions, we stated the fact that the convergents are the best rational approximations in the sense that any better rational approximation has a larger denominator.
	www.math.rutgers.edu /~sasar/Math356/?cd701.txt (261 words)

	Reverse Greed for Unit Fractions
		I suspect the reason I never thought of it before is because I tend to think in "greedy" ways, i.e., when trying to split up a fraction 7/13 my first instinct is to subtract 1/2 and then try to squeeze smaller unit fractions into the remainder.
		(This is even better than what David calls the Grouped Continued Fraction Method, which is a quite complicated algorithm, and gives 7/15 = 3' + 9' + 45'.) All in all, this new method seems to consistently produce representations that are very close to the spirit of the historical Egyptian unit fractions.
		It's more likely that their expansions tended to have pairs (or triples) of denominators with common factors because they produced them by multiplying binomials (or trinomials) of smaller expansions.
	www.mathpages.com /home/kmath150.htm (1059 words)

	Integer factorization - Wikipedia, the free encyclopedia
		In number theory, integer factorization is the process of breaking down a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer.
		When the numbers are very large, no efficient integer factorization algorithm is published; a recent effort which factored a 200 digit number (RSA-200) took eighteen months and used over half a century of computer time.
		Given the state of the art as of 2006, the hardest instances of these problems are those where the factors are two randomly-chosen prime numbers of about the same size.
	en.wikipedia.org /wiki/Prime_factorization (1214 words)

	Java programs written by Dario Alejandro Alpern
		Factorization using the Elliptic Curve Method: Applet that can be used to find 20- or 30-digit factors of numbers or numerical expressions up to 10000 digits long.
		Gaussian Integer Factorization applet: Finds the factors of complex numbers of the form a+bi where a and b are integers.
		Continued fraction calculator: This calculator can find the continued fraction expansions of rational numbers and quadratic irrationalities.
	www.alpertron.com.ar /JAVAPROG.HTM (352 words)

	NFSNET
		At its core, as with the continued fraction method and the quadratic sieve, is the goal to find a congruence of squares.
		As of November 2002, the record-holding SNFS factorization is the 233-digit factorization of 2^773+1 and the record-holding GNFS factorization is the 158-digit factorization of a divisor of 2^953+1.
		A solid understanding of the continued fraction method (CFRAC), the quadratic sieve and QS variants is a very good start.
	www.nfsnet.org /faq-nfs.html (510 words)

	[FactInt] 3 The Routines for Specific Factorization Methods
		is a continued fraction approximation for the square root of
		as possible which factor completely over a chosen factor base (a list of small primes) or with only one factor not in the factor base.
		For continuing the factorization process in another session, you will have to write this record to a file.
	www-gap.dcs.st-and.ac.uk /oldsite/pkg/factint/htm/CHAP003.htm (1358 words)

	Egyptian Fractions
		Best of Mathnerds: the magnificent seven asks for seven-term Egyptian fraction decompositions of 1, and describes a method for finding decompositions of any fraction using a method based on Farey sequences (essentially equivalent to the continued fraction method).
		minimizing denominators in unit fraction expansions, minimizing the max denominator in a t-term expansion of 1, odd greedy stubbornness, irrationality of quadratic sums, and wagon trains and sticky wickets
		Terrance Nevin uses greedy Egyptian fraction methods as a basis for investigating the dimensions of the Egyptian pyramids.
	www.ics.uci.edu /~eppstein/numth/egypt (1149 words)

	Catalogue of RPN large integer calculator functions
		Continued fraction representation of rational a ⁄ b, truncated at
		The continued fraction expansion of a ⁄ r, with r² ≡ −1 (mod a) is also given.
		mappings are continued for 2^16 iterations each, until a factor is found.
	largeint.sourceforge.net /RPNCalc.htm (1110 words)

	On Fast Computation of Continued Fractions - E-gecio-glu, Koc, Rifa, Coma (ResearchIndex)
		The algorithm is based on matrix representation of continued fractions, due to MilneThomson.
		Continued fractions can be used to calculate quadratic surds, solutions of second order linear recurrences, and various other...
		Lu Factorization And Parallel Evaluation Of Continued Fractions - Egecioglu
	citeseer.ist.psu.edu /egecioglu91fast.html (390 words)

	Amazon.com: "continued fraction method": Key Phrase page (Site not responding. Last check: 2007-10-11)
		The continued fraction method approximates a real number by a rational number with low denominator, and is related to the Euclidean algorithm.
		Suppose that a and b are two integers whose gcd is d and we wish to solve ax - by...
		Suppose that a amid b are two integers whose gel is rl and we wish to solve...
	www.amazon.com /phrase/continued-fraction-method (586 words)

	Catalogue of GP/PARI Functions: Arithmetic functions
		If x is a vector or a matrix, the factoring is done componentwise (hence the result is a vector or matrix of two-column matrices).
		In particular, the factors of rational polynomials will have integer coefficients, and the content of a polynomial or rational function is discarded and not included in the factorization.
		The algorithm fails if one of the pseudo-prime factors is not prime, which is exceedingly unlikely (and well worth a bug report).
	pari.math.u-bordeaux.fr /dochtml/html/Arithmetic_functions.html (5134 words)

	Factorization
		However, calculating such things as [5/8] and [5/16], we see that there is still a correspondence between the matrix factorization and the continued fraction factorization.
		in which case the matrix factorization does agree with the factorization given in section 2.
		This leads to the numerous cases that must be considered when considering continued fraction factorizations.
	math.arizona.edu /~rta/003/canez.santiago/node6.html (153 words)

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