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 . The golden ratio (denoted when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm .

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

(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).

Pisot constants give rise to almost integers. For example, the larger the power to which is taken, the closer , where is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).

The powers of 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 (Sloane's A086106) of

(2)

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

(3)

for , 2, 3, ...,

(4)

for , 5, 7, ..., and

(5)

for , 5, 7, ... are Pisot numbers.

All the Pisot numbers less than 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
1.3802775691	A086106	4	1 0 0
1.6216584885		16	1 2 2 1 0 0 1 2 2 1
1.8374664495		20	1 0 1 0 1 0 1 0 0 1 0 1 0 1

Pisot numbers originally arose in the consideration of

(6)

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

Pisot (1938) subsequently proved the fact that if is chosen such that there exists a for which the series

(7)

converges, then is an algebraic integer whose conjugates all (except for itself) have modulus , and is an algebraic integer of the field . 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 , there always exists a number such that and the following inequality is satisfied:

(8)

Portions of this entry contributed by David Terr

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Terr, David and Weisstein, Eric W. "Pisot Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PisotNumber.html