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	PlanetMath: integer factorization
		The factorization of a positive integer is unique (this is the fundamental theorem of arithmetic).
		The term “factorization” is often used to refer to the actual process of determining the prime factors.
		This is version 4 of integer factorization, born on 2007-02-02, modified 2007-07-11.
	planetmath.org /encyclopedia/IntegerFactorization.html (331 words)

	Factorization - Encyclopedia, History, Geography and Biography
		In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original.
		The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials.
		Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.
	www.arikah.com /encyclopedia/Factorisation (1252 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).
		However, unless a prior sequence of small primes is available, it's simpler to use a sequence of trial divisors (starting with 2) which includes all primes without ruling out composite numbers.
		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 (2791 words)

	InterMath / Dictionary / Description
		A prime factor has no factors other than itself and one, so there's no need to continue branching.) The prime factors are located at the "leaves" of the tree.
		The prime factors located on the "leaves" form the prime factorization of the composite number you started with.
		The prime factorization of 220 is found by pulling the "leaves" (prime numbers) off the branches.
	intermath.coe.uga.edu /dictnary/descript.asp?termID=272 (237 words)

	Prime Factorization
		It is simply assumed that 1 is a factor of all whole numbers (which it is) but it would be redundant to include it in every solution for primes, so we don't.
		Prime factors are the factors of a number that also happen to be prime.
		But we know that no prime numbers that are smaller than 17 are factors of 293, therefore there is no chance that a larger number could be multiplied by one of them to give you 293.
	www.mathmojo.com /interestinglessons/prime_factorization/pretty_good_guide_to_prime_factorization.html (2655 words)

	RSA Laboratories - 2.3.3 What is the factoring problem?
		Factoring is the act of splitting an integer into a set of smaller integers (factors) which, when multiplied together, form the original integer.
		Prime factorization requires splitting an integer into factors that are prime numbers; every integer has a unique prime factorization.
		What is true in general is that a number with large prime factors is more difficult to factor than a number with small prime factors (still, the running time of some factoring algorithms depends on the size of the number only and not on the size of its prime factors).
	www.rsa.com /rsalabs/node.asp?id=2189 (408 words)

	Prime Factorization
		A good way to find the prime factorization of a number is to see if it can be divided by these small primes.
		Again, the prime factorization of 24 is 2 × 2 × 2 × 3.
		Factoring polynomials -- a second year algebra topic, which requires that you be able to factor whole numbers.
	2000clicks.com /MathHelp/NumberPrimeFactorization.htm (585 words)

	SparkNotes: Whole Numbers: Prime Factorization, the Greatest Common Factor, and the Least Common Multiple
		Therefore, the prime factorization of 1,575 is 3×3×5×5×7.
		For example, the prime factorization of 40 is 2×2×2×5 and the prime factorization of 21 is 3×7.
		Since 40 and 21 have no common prime factors, they are said to be relatively prime, and their greatest common factor is 1.
	www.sparknotes.com /math/prealgebra/wholenumbers/section4.rhtml (723 words)

	The Prime Glossary: wheel factorization
		To see if a number is prime via trial division (or to find its prime factors), we divide by all of the primes less than (or equal to) its square root.
		Rather than divide by just the primes, it is sometimes more practical to divide by 2, 3, and 5; then divide by all the numbers congruent to 1, 7, 11, 13, 17, 19, 23, and 29 modulo 30--again stopping when you reach the square root.
		The density of primes decreases as the integers increase in size (see the prime number theorem), so when we apply these same wheels to a list of large integers, almost all of those that are not removed by the wheel are composite.
	primes.utm.edu /glossary/page.php?sort=WheelFactorization (488 words)

	Prime Factorization and Decomposition, Composite Numbers
		Recognition of prime factors and the prime decomposition of whole numbers aid calculations of LCM, GCD and LCD in arithmetic with whole numbers and fractions.
		A whole number is not prime if is a proper multiple of 2, 3, 5, 7 or 11, or equivalently if it remainder, modulo these small primes is zero.
		Given any finite sequence of prime numbers, their product plus one is a prime or is a multiple of a prime not in the sequence.
	whyslopes.com /Number_Theory/Decimal_Prime_Factorization_Methods.html (1073 words)

	Math Forum - Ask Dr. Math
		Prime numbers are numbers that you can't divide without getting a fraction, unless you feel like dividing them by themselves or by 1.
		All prime factorization is, is taking a composite number and splitting it up into the little numbers that it's made up of until you can't split it up any more.
		If you draw the smaller pieces (called the factors) in an upside down tree, like I just did, you can go back and collect all the pieces at the ends of the branches, and they will be the "prime factors" of the number.
	mathforum.org /library/drmath/view/58528.html (827 words)

	Prime Factorization
		A good way to find the prime factorization of a number is to see if it can be divided by these small primes.
		Again, the prime factorization of 24 is 2 × 2 × 2 × 3.
		Factoring polynomials -- a second year algebra topic, which requires that you be able to factor whole numbers.
	mcraefamily.com /MathHelp/BasicNumberPrimeFactorization.htm (528 words)

	Factorization Summary
		Factoring large numbers was once mainly of interest to mathematicians, but today factoring is the basis of the security codes used by computers in military codes and in protecting financial transactions.
		The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials.
		Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.
	www.bookrags.com /Factorization (2193 words)

	Prime Factorization (Site not responding. Last check: )
		Prime numbers are numbers that have only two factors; 1 and themselves.
		Prime factors are factors of a number that are, themselves, prime numbers.
		Prime factorization is the breaking down of a number into it's prime factors.
	www.lc.capellauniversity.edu /~121064/7503/tutorial/prime.htm (56 words)

	Proofs Regarding Primorial Patterns
		Observations are made regarding the pattern of the composite numbers that have a particular prime factor for their lowest prime factor.
		That there are three numbers that have 2 for their lowest prime factor makes sense because we multiplied the previous primorial by 3 and that primorial interval had one number that had 2 for its lowest prime factor.
		Ignoring the trivial case where one of those is 1, those two numbers individually are either prime or have a prime that is higher than 3 for their lowest prime factor.
	www.primenace.com /papers/math/PrimorialPatternProofs.htm (1431 words)

	Prime numbers - A complete course in arithmetic
		With the exception of 2, then, -- which is the only even prime -- a prime number is a kind of odd number.
		If it is prime, then we have found a prime that is not on the list, and the theorem is proved.
		Therefore p is not one of the primes of N, which is to say, p is not a prime on the list.
	www.themathpage.com /arith/prime-numbers.htm (1991 words)

	Factoring Numbers
		This accidental over-duplication of factors is another reason why the prime factorization is often best: it avoids counting any factor too many times.
		So it's best to stick to the prime factorization, even if the problem doesn't require it, in order to avoid either omitting a factor or else over-duplicating one.
		Since the square of your prime is bigger than the number, then a bigger prime must have as its remainder a smaller number than your prime.
	www.purplemath.com /modules/factnumb.htm (685 words)

	Factorizer: A Factorization and Prime Number Program
		Factorizer is a Windows program to find factors of numbers up to 2,147,483,646 and to find primes, pairs of primes and Palmen colors of numbers.
		(5) to count (without displaying) the prime numbers in a range of numbers.
		Following accepted mathematical convention, the number 1 is not held to be a prime number.
	www.hermetic.ch /factors/factors.htm (428 words)

	Prime Factorization - Math
		Prime Factorization is used to find the Least Common Multiple and the Greatest Common Factor.
		When a composite number is factored using only prime numbers such as 2 x 3 x 5 = 30, it is referred to as prime factorization.
		If the answer is a prime number, divide the number by itself to get one and you are finished; include all prime numbers in the factorization.
	www.bellaonline.com /articles/art22022.asp (512 words)

	Prime Factorization on the Abacus
		If it is, place this first prime number on the extreme left of the abacus and divide the dividend by the prime.
		Continue (putting each new prime that is a factor in the column to the right of the last factor and replacing the quotient with a new dividend) until you arrive at a quotient of 1.
		If the number to be prime factored is quite large, you may wish to use two abaci - one for the dividend and one for the factors.
	www.tsbvi.edu /math/primefactor.htm (643 words)

	Prime Factorization
		This skill is indispensable for reducing fractions, finding common factors in algebra, and to help with ‘blowing up’ fractions when you change denominators.
		The basic idea is to list all the factors (other than 1 and the original number) for each number under consideration.
		For best practice results (and also to get familiar with higher factors) make yourself a list of all the numbers up to some decent size (like 50 or 100) and find the factors for all of them in order.
	www.msu.edu /user/mcwillia/prime.htm (301 words)

	Factorization
		After that, if a composite factor of at least 25 digits remains, MPQS is used; it is the best method available for factoring integers of more than about 40 decimal digits especially for products of two primes of roughly equal size.
		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.
		All the divisors in F_i are square factors, and, for any i and j, the divisors in G_i and G_j are either equal or are pairwise coprime.
	www.umich.edu /~gpcc/scs/magma/text540.htm (2378 words)

	Prime number - Conservapedia
		A prime number is a natural number that is divisible by only 1 and itself.
		The probability that a random integer N is prime is roughly 1/ln(N), where ln(N) is the natural logarithm (base e) of N. This is a formulation of a more general statement known as the prime number theorem.
		The security of public-key cryptography schemes such as RSA depend on the difficulty of prime factorization, if a pattern in the distribution of prime numbers is discovered, then such cryptographic schemes could become vulnerable to attack.
	www.conservapedia.com /Prime_number (779 words)

	Algebra — Prime Factorization - For Dummies
		A prime factorization of a number is a unique (only one possible) listing of all the prime numbers whose product gives you that number.
		The prime factorization of a prime number is just that number, because 1 isn't prime.
		To find the prime factorization of a number, you can start by thinking of any two numbers whose product is that number, and then take each of those two numbers and think of two numbers whose product is each of them, and so on until only prime numbers are being multiplied.
	www.dummies.com /WileyCDA/DummiesTip/id-3006,subcat-MATH.html (200 words)

	Prime and Composite Numbers/Prime Factorization - Math Lesson Plan, Thematic Unit, Activity, Worksheet, or Teaching Idea
		She will also introduce prime factorization of composite numbers.
		She will model prime factorization, using the colorful poster board.
		If it is composite, they will demonstrate prime factorization, again using their construction squares.
	www.lessonplanspage.com /MathCompositeNumbersPrimeFactorization5.htm (570 words)

	Prime Factorization Chip - Home Page - Prime Numbers: are they errors in multiplication???
		Prime factors are the basic building blocks of any positive integer.
		Prime factorization of a number is the process to find that unique sequence of prime numbers.
		Once we know the prime factors of two or more numbers we can quickly find their Greatest Common Divisor and Least Common Multiple.The basic motivation for the selection of this title is the underlying importance of prime factors of numbers.
	www.eecs.tufts.edu /~nauman/primefactor (362 words)

	Science and Reason: Factorization of prime ideals in extension fields
		First, we'll look at how a prime ideal of one ring of algebraic integers factors into multiple prime ideals in the ring of integers of a larger field.
		If we start with a prime ideal P of the integers of an extension of ℚ, then P∩ℤ is a prime ideal (p) of ℤ for some rational prime p, and P lies above (p).
		Primes that ramify tend to make life more complicated, but fortunately there are only finitely many for any particular extension.
	scienceandreason.blogspot.com /2008/04/factorization-of-prime-ideals-in.html (1505 words)

	Factorization
		In a broad sense, 0 can be factorized, as 01, 02, 03, 05, etc. But none of these are prime factorizations, because 0 is not a prime but is involved.
		We feel that 25 should be the ``only'' prime* factorization of 10, and things like 125 and (-1)(-1)2*5 should be merely insignificant, superfluous variations.
		Having settled the issues of 0 and units, it can be proved that numbers can indeed be factorized into primes, and the prime factorization of a number is fairly unique.
	www.vex.net /~trebla/numbertheory/factor.html (643 words)

	Number and Operations Session 5, Part C:Factors
		Some numbers factor into two factors only, while others may have two factors, one or both of which can be factored further.
		An important distinction can be made between the terms "factor" and "prime factor." By factors, we mean all the factors of a number.
		Use the Interactive Activity to explore the factors and prime factorizations of the numbers from 2 to 36.
	www.learner.org /channel/courses/learningmath/number/session5/part_c/index.html (487 words)

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