Passive integrator circuit: Difference between revisions
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{{distinguish|Binomial distribution}} | |||
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]] | |||
Latest revision as of 19:35, 3 March 2013
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 , with and . However, since is always divisible by , when studying the numbers generated from the version with the negative sign, they are usually divided by first. Binomial numbers formed this way form Lucas sequences. Specifically:
Binomial numbers are a generalization of a Cunningham numbers, and it will be seen that the Cunningham numbers are Binomial numbers where . Other subsets of the Binomial numbers are the Mersenne numbers and the Repunits.
Factorization
The main reason for studying these numbers is to obtain their factorizations. Aside from algebraic factors, which are obtained by 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 they do not factorize with ) which occur seemingly at random, and it is these which the number theorist is looking for.
Some Binomial numbers' underlying binomials have Aurifeuillian factorizations,[1] which can assist in finding prime factors. Cyclotomic polynomials are also helpful in finding factorizations.[2]
The amount of work required in searching for a factor is considerably reduced by applying Legendre's theorem.[3] This theorem states that all factors of a binomial number are of the form if is even or 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
References
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