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 .
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).
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:
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