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 
(OEIS 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, ...
(OEIS A051016), while those for which it is
closer to 1 are 2, 9, 10, 13, 15, 16, 18, 20, 21, 23, ... (OEIS A051017).
Siegel also identified the second smallest Pisot numbers as the positive root 
(OEIS 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 | OEIS | 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)
|
for 
a Pisot number."