\alpha>1
Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be a reciprocal polynomial. This implies that
1/\alpha
Every Salem number is a Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).
The smallest known Salem number is the largest real root of Lehmer's polynomial (named after Derrick Henry Lehmer)
P(x)=x10+x9-x7-x6-x5-x4-x3+x+1,
which is about
x=1.17628
Lehmer's polynomial is a factor of the shorter degree-12 polynomial,
Q(x)=x12-x7-x6-x5+1,
all twelve roots of which satisfy the relation[2]
x630-1=
(x315-1)(x210-1)(x126-1)2(x90-1)(x3-1)3(x2-1)5(x-1)3 | |
(x35-1)(x15-1)2(x14-1)2(x5-1)6x68 |
Salem numbers can be constructed from Pisot–Vijayaraghavan numbers. To recall, the smallest of the latter is the unique real root of the cubic polynomial,
x3-x-1,
known as the plastic ratio and approximately equal to 1.324718. This can be used to generate a family of Salem numbers including the smallest one found so far. The general approach is to take the minimal polynomial
P(x)
P*(x)
xnP(x)=\pmP*(x)
n
xn(x3-x-1)=-(x3+x2-1)
then for
n=8
(x-1)(x10+x9-x7-x6-x5-x4-x3+x+1)=0
where the decic is Lehmer's polynomial. Using higher
n
n
x(x3-x-1)1/n=\pm(x3+x2-1)1/n
so as
n
x
x3-x-1=0
x
xn(x4-x3-1)=-(x4+x-1),
which for
n=7
(x-1)(x10-x6-x5-x4+1)=0,
a decic not generated in the previous and has the root
x=1.216391\ldots
n\toinfty
x4-x3-1=0
. Peter Borwein . Computational Excursions in Analysis and Number Theory . CMS Books in Mathematics . . 2002 . 0-387-95444-9 . 1020.12001 . Chap. 3.
. Raphaël Salem . Algebraic numbers and Fourier analysis . Heath mathematical monographs . Boston, MA . . 1963 . 0126.07802 .