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=> Algebraic Curves-Mordell Curve
=> Algebraic Curves-Ochoa Curve
=> Algebraic Integer
=> Algebraic Number
=> Algebraic Number Theory
=> Chebotarev Density Theorem
=> Class Field
=> Cyclotomic Field
=> Dedekind Ring
=> Fractional Ideal
=> Global Field
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=> Number Field Signature
=> Picard Group
=> Pisot Number
=> Weyl Sum
=> Casting Out Nines
=> A-Sequence
=> Anomalous Cancellation
=> Archimedes' Axiom
=> B2-Sequence
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=> Egyptian Fraction
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=> Erdős-Turán Conjecture
=> Eye of Horus Fraction
=> Farey Sequence
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=> One-Seventh Ellipse
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Ziyaretçi defteri
 

Pisot Number

A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to +/-1. The golden ratio phi (denoted theta_0 when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm -1.

The smallest Pisot number is given by the positive root theta_1=1.324717957... (Sloane's A060006) of

 x^3-x-1=0,
(1)

known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).

PisotConstant

Pisot constants give rise to almost integers. For example, the larger the power to which theta_1 is taken, the closer theta_1^n-|_theta_1^n_|, where |_x_| is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example theta_1^(27369) is within 1.18463×10^(-1671) of an integer (Trott 2004, pp. 8-9).

The powers of theta_1 for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (Sloane's A051016), while those for which it is closer to 1 are 2, 9, 10, 13, 15, 16, 18, 20, 21, 23, ... (Sloane's A051017).

Siegel also identified the second smallest Pisot numbers as the positive root theta_2=1.38027756... (Sloane's A086106) of

 x^4-x^3-1=0,
(2)

showed that theta_1 and theta_2 are isolated, and showed that the positive roots of each polynomial

 x^n(x^2-x-1)+x^2-1
(3)

for n=1, 2, 3, ...,

 x^n-(x^(n+1)-1)/(x^2-1)
(4)

for n=3, 5, 7, ..., and

 x^n-(x^(n-1)-1)/(x-1)
(5)

for n=3, 5, 7, ... are Pisot numbers.

All the Pisot numbers less than phi are known (Dufresnoy and Pisot 1955). Some small Pisot numbers and their polynomials are given in the following table. The latter two entries are from Boyd (1977).

number Sloane order polynomial coefficients
1.3247179572 A060006 3 1 0 -1 -1
1.3802775691 A086106 4 1 -1 0 0 -1
1.6216584885   16 1 -2 2 -3 2 -2 1 0 0 1 -1 2 -2 2 -2 1 -1
1.8374664495   20 1 -2 0 1 -1 0 1 -1 0 1 0 -1 0 1 -1 0 1 -1 0 1 -1

Pisot numbers originally arose in the consideration of

 frac(x)=x-|_x_|
(6)

where frac(x) denotes the fractional part of x and |_x_| is the floor function. Letting theta be a number greater than 1 and lambda a positive number, for a given lambda, the sequence of numbers frac(lambdatheta^n) for n=1, 2, ... is an equidistributed sequence in the interval (0, 1) when theta does not belong to a lambda-dependent exceptional set S of measure zero (Koksma 1935). Pisot (1938) and Vijayaraghavan (1941) independently studied the exceptional values of theta, and Salem (1943) proposed calling such values Pisot-Vijayaraghavan numbers.

Pisot (1938) subsequently proved the fact that if theta is chosen such that there exists a lambda!=0 for which the series

 sum_(n=0)^inftysin^2(pilambdatheta^n)
(7)

converges, then theta is an algebraic integer whose conjugates all (except for itself) have modulus <1, and lambda is an algebraic integer of the field K(theta). Vijayaraghavan (1940) proved that the set of Pisot numbers has infinitely many limit points. Salem (1944) proved that the set of Pisot numbers is closed. The proof of this theorem is based on the lemma that for a Pisot number theta, there always exists a number lambda such that 1<=lambda<theta and the following inequality is satisfied:

 sum_(n=0)^inftysin^2(pilambdatheta^n)<=(pi^2(2theta+1)^2)/((theta-1)^2).
(8)
 

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