Minimal polynomials of the Salem numbers less than 1.3 - NumberDB

Minimal polynomials of the Salem numbers less than 1.3

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Polynomials

coefficients

$m_a(x)$

1, 1, 0, -1, -1, -1:

x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1

comment: This is Lehmer's polynomial; its real root greater than $1$ is Lehmer's number, the smallest known Salem number.

1, -1, 1, -1, 0, 0, -1, 1, -1, 1:

x^18 - x^17 + x^16 - x^15 - x^12 + x^11 - x^10 + x^9 - x^8 + x^7 - x^6 - x^3 + x^2 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, -1, -1, 0, 0, 1:

x^14 - x^11 - x^10 + x^7 - x^4 - x^3 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, 0, 0, -1:

x^14 - x^12 - x^7 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, -1, -1:

x^10 - x^6 - x^5 - x^4 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, 0, 0, -1, 1:

x^18 - x^17 - x^10 + x^9 - x^8 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, -1, 0, -1:

x^10 - x^7 - x^5 - x^3 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, -1, 1, 0, 0, -1, 1:

x^20 - x^19 - x^15 + x^14 - x^11 + x^10 - x^9 + x^6 - x^5 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, -1, 0, 0, 0, 1, 1, 0, -1, -1:

x^22 - x^20 - x^19 + x^15 + x^14 - x^12 - x^11 - x^10 + x^8 + x^7 - x^3 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, 0, 0, -1:

x^16 - x^15 - x^8 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1:

x^26 - x^24 - x^21 - x^18 + x^16 + x^13 + x^10 - x^8 - x^5 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 1, -1, 0, 0, -1:

x^12 - x^11 + x^10 - x^9 - x^6 - x^3 + x^2 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, 0, 0, -1, -1, -1, -1:

x^18 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0:

x^20 - x^18 - x^15 - x^5 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, -1, 0, 1, 0, -1:

x^14 - x^12 - x^11 + x^9 - x^7 + x^5 - x^3 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, -1, 1, 0, 0, 0, -1:

x^18 - x^17 - x^14 + x^13 - x^9 + x^5 - x^4 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, -1, 1, 0, -1, 1, -1, 0, 1, -1:

x^24 - x^23 - x^20 + x^19 - x^17 + x^16 - x^15 + x^13 - x^12 + x^11 - x^9 + x^8 - x^7 + x^5 - x^4 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, -1, 1, 0, 0, 0, -1, 1, -1, 1:

x^22 - x^21 - x^19 + x^18 - x^14 + x^13 - x^12 + x^11 - x^10 + x^9 - x^8 + x^4 - x^3 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, -1:

x^10 - x^8 - x^5 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1:

x^26 - x^25 - x^20 + x^13 - x^6 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, -1, 1:

x^14 - x^13 - x^8 + x^7 - x^6 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1:

x^22 - x^21 - x^20 + x^19 - x^13 + x^11 - x^9 + x^3 - x^2 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, -1, -1:

x^8 - x^5 - x^4 - x^3 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, -1, -1:

x^26 - x^20 - x^19 - x^18 - x^17 - x^16 - x^15 - x^14 - x^13 - x^12 - x^11 - x^10 - x^9 - x^8 - x^7 - x^6 + 1

comment: $\tau$ is the real root greater than $1$.

1, -2, 2, -2, 2, -2, 1, 0, -1, 1, -1:

x^20 - 2*x^19 + 2*x^18 - 2*x^17 + 2*x^16 - 2*x^15 + x^14 - x^12 + x^11 - x^10 + x^9 - x^8 + x^6 - 2*x^5 + 2*x^4 - 2*x^3 + 2*x^2 - 2*x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, -1, 0, -1, -1, 0, -1:

x^18 - x^14 - x^12 - x^11 - x^9 - x^7 - x^6 - x^4 + 1

comment: $\tau$ is the real root greater than $1$.

1, -2, 1, 1, -2, 1, 0, 0, -1, 1, 0, -1, 1, -1:

x^26 - 2*x^25 + x^24 + x^23 - 2*x^22 + x^21 - x^18 + x^17 - x^15 + x^14 - x^13 + x^12 - x^11 + x^9 - x^8 + x^5 - 2*x^4 + x^3 + x^2 - 2*x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, 0, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 1:

x^30 - x^25 - x^24 - x^23 - x^22 - x^21 - x^20 + x^15 - x^10 - x^9 - x^8 - x^7 - x^6 - x^5 + 1

comment: $\tau$ is the real root greater than $1$.

1, -2, 2, -2, 1, 0, -1, 2, -2, 1, 0, -1, 1, -1, 1, -1:

x^30 - 2*x^29 + 2*x^28 - 2*x^27 + x^26 - x^24 + 2*x^23 - 2*x^22 + x^21 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^9 - 2*x^8 + 2*x^7 - x^6 + x^4 - 2*x^3 + 2*x^2 - 2*x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1:

x^30 - x^29 - x^22 - x^18 - x^15 - x^12 - x^8 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, -1, 0, 0, 0, 1, 0, -1, -1, 0, 1, 1:

x^26 - x^24 - x^23 + x^19 - x^17 - x^16 + x^14 + x^13 + x^12 - x^10 - x^9 + x^7 - x^3 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1:

x^44 - x^43 - x^37 - x^33 + x^25 + x^22 + x^19 - x^11 - x^7 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, 1, 0, -1:

x^30 - x^28 - x^25 - x^24 + x^20 + x^17 - x^15 + x^13 + x^10 - x^6 - x^5 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1:

x^34 - x^33 - x^30 + x^29 - x^28 + x^26 - x^25 + x^24 - x^22 + x^21 - x^20 + x^18 - x^17 + x^16 - x^14 + x^13 - x^12 + x^10 - x^9 + x^8 - x^6 + x^5 - x^4 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -2, 2, -2, 2, -2, 2, -3, 3, -3:

x^18 - 2*x^17 + 2*x^16 - 2*x^15 + 2*x^14 - 2*x^13 + 2*x^12 - 3*x^11 + 3*x^10 - 3*x^9 + 3*x^8 - 3*x^7 + 2*x^6 - 2*x^5 + 2*x^4 - 2*x^3 + 2*x^2 - 2*x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1:

x^26 - x^25 - x^22 + x^21 - x^20 + x^18 - x^17 + x^16 - x^14 + x^13 - x^12 + x^10 - x^9 + x^8 - x^6 + x^5 - x^4 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0:

x^24 - x^23 - x^18 - x^6 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, 0, 0, -1, 0, 0, -1, 0, 1:

x^20 - x^18 - x^15 - x^12 + x^10 - x^8 - x^5 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 1, 0, 1:

x^40 - x^37 - x^35 - x^33 - x^31 - x^29 + x^26 + x^24 + x^22 + x^20 + x^18 + x^16 + x^14 - x^11 - x^9 - x^7 - x^5 - x^3 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1:

x^46 - x^42 - x^41 - x^40 - x^39 + x^25 + x^24 + x^23 + x^22 + x^21 - x^7 - x^6 - x^5 - x^4 + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, -1, -1, 0, 1:

x^10 - x^8 - x^7 + x^5 - x^3 - x^2 + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, -1, 1, -1, 0, 1, -1:

x^18 - x^17 - x^14 + x^13 - x^12 + x^10 - x^9 + x^8 - x^6 + x^5 - x^4 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, -1, 0, 1, 0, 1, -2, 0, 0, 1, 1, -1, -1, -1, 1, 1:

x^34 - x^33 - x^31 + x^29 + x^27 - 2*x^26 + x^23 + x^22 - x^21 - x^20 - x^19 + x^18 + x^17 + x^16 - x^15 - x^14 - x^13 + x^12 + x^11 - 2*x^8 + x^7 + x^5 - x^3 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1:

x^22 - x^21 - x^17 + x^11 - x^5 - x + 1

comment: $\tau$ is the real root greater than $1$.

1, 0, 0, 0, -1, -1, -1, -1, -1, 0, 0, 0, 1, 1, 1:

x^28 - x^24 - x^23 - x^22 - x^21 - x^20 + x^16 + x^15 + x^14 + x^13 + x^12 - x^8 - x^7 - x^6 - x^5 - x^4 + 1

comment: $\tau$ is the real root greater than $1$.

1, 1, 0, -1, -2, -2, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1:

x^36 + x^35 - x^33 - 2*x^32 - 2*x^31 - x^30 + x^28 + x^27 - x^25 - x^24 + x^22 + x^21 - x^19 - x^18 - x^17 + x^15 + x^14 - x^12 - x^11 + x^9 + x^8 - x^6 - 2*x^5 - 2*x^4 - x^3 + x + 1

comment: $\tau$ is the real root greater than $1$.

1, -1, -1, 0, 2, 0, -2, -1, 2, 2, -2, -2, 0, 3:

x^26 - x^25 - x^24 + 2*x^22 - 2*x^20 - x^19 + 2*x^18 + 2*x^17 - 2*x^16 - 2*x^15 + 3*x^13 - 2*x^11 - 2*x^10 + 2*x^9 + 2*x^8 - x^7 - 2*x^6 + 2*x^4 - x^2 - x + 1

comment: $\tau$ is the real root greater than $1$.

Definition

A Salem number is an algebraic integer $\tau>1$ whose other conjugates have absolute value at most $1$, with at least one on the unit circle [4]. The table stores the minimal polynomial $m_a(x)\in\mathbb{Z}[x]$ of the Salem numbers $\tau<1.3$ listed here.

Parameters

coefficients

— leading coefficients of the minimal polynomial (the coefficients $a_0,\ldots,a_{d/2}$ of $x^d,\ldots,x^{d/2}$ in the minimal polynomial of a Salem number of even degree $d$)

Formulas

(1)

For coefficients $a_0,a_1,\ldots,a_{d/2}$, the polynomial is $m_a(x)=a_0x^d+a_1x^{d-1}+\cdots+a_{d/2}x^{d/2}+\cdots+a_1x+a_0$.

Comments

(2)

Rows are ordered by increasing Salem root $\tau$. The coefficient tuple $a=(a_0,\ldots,a_{d/2})$ gives the coefficients of $x^d,\ldots,x^{d/2}$ in the monic reciprocal polynomial, as in Formula (1). The same tuple indexes the corresponding root in Salem numbers less than $1.3$, where $\tau$ is given to $100$ digits.

Programs

(P1)

Sage

R.<x> = ZZ[]
half = [1, 1, 0, -1, -1, -1]
d = 2*(len(half) - 1)
sum(half[i]*x^(d - i) for i in range(len(half))) + sum(half[i]*x^i for i in range(len(half) - 1))

References

[1]

J.-M. Sac-Épée, Salem numbers less than $49/37$, 2025. (arXiv) (doi)

[2]

M. J. Mossinghoff, G. Rhin and Q. Wu, Minimal Mahler measures, Experimental Mathematics 17 (2008), no. 4, 451-458. (doi) (MR)

[3]

M. J. Mossinghoff, Polynomials with small Mahler measure, Mathematics of Computation 67 (1998), no. 224, 1697-1705, S11-S14. (doi)

Links

[4]

Wikipedia: Salem number

[5]

Mossinghoff: Small Salem Numbers

Similar tables

Salem numbers less than $1.3$ — stores the Salem root $\tau>1$ of each polynomial $m_a(x)$ held here

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds the minimal polynomials of the 47 known Salem numbers below $1.3$ in Mossinghoff's list [5]. The list is proved complete for degree at most $44$ [2]. The one degree-$46$ entry lies outside that range, and a later random-sampling search rediscovered all 47 and found no others below $1.3$ [1])

How they were obtained:

The generator parses the exact coefficient half-lists in Mossinghoff's archived table [5]. It constructs the reciprocal polynomial in Formula (1), checks irreducibility over $\mathbb{Q}$, checks the Salem root pattern, and stores the resulting monic integer polynomial exactly.

more

It also verifies that the real root greater than $1$ lies in the corresponding interval in the Salem-number table.