PlanetMath: Salem number

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Salem number

(Definition)

Salem number is a real algebraic integer $\alpha>1$ whose algebraic conjugates all lie in the unit disk $\{\,z\in\mathbb{C} \,\big\vert\, \vert z\vert\leq 1\,\}$ with at least one on the unit circle $\{\,z\in\mathbb{C}\,\big\vert\,\vert z\vert= 1\,\}$ .

Powers of a Salem number $\alpha^n\ (n=1,2,\dotsc)$ are everywhere dense modulo , but are not uniformly distributed modulo .

The smallest known Salem number is the largest positive root of

$\displaystyle \alpha^{10}+\alpha^9-\alpha^7-\alpha^6-\alpha^5-\alpha^4-\alpha^3+\alpha+1=0.$

"Salem number" is owned by bbukh.

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Cross-references: positive root, uniformly distributed, everywhere dense, unit circle, unit disk, algebraic conjugates, algebraic integer, real

This is version 3 of Salem number, born on 2003-05-26, modified 2003-10-02.
Object id is 4298, canonical name is SalemNumber.
Accessed 1469 times total.

Classification:

AMS MSC:	11R06 (Number theory :: Algebraic number theory: global fields :: PV-numbers and generalizations; other special algebraic numbers)
	11J71 (Number theory :: Diophantine approximation, transcendental number theory :: Distribution modulo one)

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