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 
(denoted 
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 
(Sloane's A060006) of
 |
(1) |
This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).
Pisot-Vijayaraghavan constants give rise to almost integers. For example, the larger the power to which is taken, the closer 
,
is the floor function, is to either 0 or 1 (Trott 2004). 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 n = 1, 2, 3, ...,
 |
(4) |
for n = 3, 5, 7, ..., and
 |
(5) |
for n = 3, 5, 7, ....
A more spectacular example is
 |
(6) |
which is within 
of an integer (Trott 2004, pp. 8-9).
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 -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
 |
(7) |
where 
denotes the fractional part of x and is the floor function. Letting 
be a number greater than 1 and 
a positive number, for a given ,
for n = 1, 2, ... is an equidistributed sequence in the interval (0, 1) when does not belong to a -dependent exceptional set S of measure zero (Koksma 1935). Pisot (1938) and Vijayaraghavan (1941) independently studied the exceptional values of ,
Pisot (1938) subsequently proved the fact that if is chosen such that there exists a 
for which the series
 |
(8) |
converges, then is an algebraic integer whose conjugates all (except for itself) have modulus 
, is an algebraic integer of the field 
., such that 
and the following inequality is satisfied:
 |
(9) |
Almost Integer, Equidistributed Sequence, Salem Constants, Weyl's Criterion
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