en>Martin Berka: Other-language articles on explicitly passive integrators

2013-03-03T17:35:23Z

Other-language articles on explicitly passive integrators

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			In [[mathematics]], specifically in [[number theory]], a '''binomial number''' is an [[integer]] which can be obtained by evaluating a [[homogeneous polynomial]] containing two terms. It is a generalization of a [[Cunningham number]].

			== Definition ==

			A '''Binomial number''' is an [[integer]] obtained by evaluating a [[homogeneous polynomial]] containing two terms, also called a [[binomial]]. The form of this binomial is <math>\scriptstyle x^n \,\pm\, y^n</math>, with <math>\scriptstyle x \, > \, y</math> and <math>\scriptstyle n \, > \, 1 </math>. However, since <math>\scriptstyle x^n \,-\, y^n</math> is always divisible by <math>\scriptstyle x \,-\, y</math>, when studying the numbers generated from the version with the negative sign, they are usually divided by <math>\scriptstyle x \,-\, y</math> first. Binomial numbers formed this way form [[Lucas sequence]]s. Specifically:

			:<math>U_n(a+b,ab)=\frac{a^n-b^n}{a-b}, \,</math> and <math>V_n(a+b,ab)=a^n+b^n \,</math>
			<!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->

			Binomial numbers are a generalization of a [[Cunningham number]]s, and it will be seen that the [[Cunningham number]]s are Binomial numbers where <math>\scriptstyle y \,=\, 1 </math>. Other subsets of the Binomial numbers are the [[Mersenne numbers]] and the [[Repunit]]s.

			==Factorization ==

			The main reason for studying these numbers is to obtain their [[factorization]]s. Aside from algebraic [[divisor\|factors]], which are obtained by [[factorization\|factoring]] the underlying [[polynomial]] ([[binomial]]) that was used to define the number, there are other [[prime factors]] (called primitive prime factors, because for a given <math>\scriptstyle x^n \,\pm\, y^n</math> they do not factorize <math>\scriptstyle x^m \,\pm\, y^m</math> with <math>\scriptstyle m \, < \, n</math>) which occur seemingly at random, and it is these which the number theorist is looking for.

			Some Binomial numbers' underlying [[binomials]] have [[Aurifeuillian factorization]]s,<ref>{{Harvard citations \| last1=Riesel \| year=1994\|loc=p. 309\|nb=yes}}</ref> which can assist in finding [[prime factor]]s. [[Cyclotomic polynomial]]s are also helpful in finding factorizations.<ref>{{Harvard citations \| last1=Riesel \| year=1994\|loc=p. 305\|nb=yes}}</ref>


			The amount of work required in searching for a factor is considerably reduced by applying Legendre's theorem.<ref>{{Harvard citations \| last1=Riesel \| year=1994\|loc=p. 165\|nb=yes}}</ref> This theorem states that all factors of a binomial number are of the form <math>\scriptstyle kn \, + \, 1</math> if <math>\scriptstyle n \,</math> is even or <math>\scriptstyle 2kn \, + \, 1</math> if it is odd.

			== Observation ==

			Some people write "binomial number" when they mean [[binomial coefficient]], but this usage is not standard and is deprecated.

			==See also==
			*[[Cunningham project]]

			==References==
			{{Reflist}}

			==External links==
			*[http://mathworld.wolfram.com/BinomialNumber.html Binomial Number at MathWorld]

			[[Category:Number theory]]

169.237.129.226: Grammar: fixed a comma splice. Two independent clauses may not be joined by a comma.

2011-03-03T21:28:30Z

Grammar: fixed a comma splice. Two independent clauses may not be joined by a comma.

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Andrew Simcox is the name his mothers and fathers gave him and he completely loves this name. What I love doing is soccer but I don't have the time lately. For many years he's been residing in Mississippi and he doesn't plan on changing it. Credit authorising is how he tends to make cash.<br><br>Also visit my web-site :: [http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ psychic readings online] phone online psychic readings ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 pusan.ac.kr]) readings, [http://Gcjcteam.org/index.php?mid=etc_video&document_srl=696611&sort_index=regdate&order_type=desc visit the following site],

Passive integrator circuit - Revision history

en>Martin Berka: Other-language articles on explicitly passive integrators

169.237.129.226: Grammar: fixed a comma splice. Two independent clauses may not be joined by a comma.