Salem Constants

Each point of the Pisot-Vijayaraghavan Constants $S$ is a Limit Point from both sides of a set $T$ known as the Salem constants (Salem 1945). The Salem constants are algebraic Integers $>1$ in which one or more of the conjugates is on the Unit Circle with the others inside (Le Lionnais 1983, p. 150). The smallest known Salem number was found by Lehmer (1933) as the largest Real Root of

$$x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1=0,$$

which is

$$\sigma_1=1.176280818\ldots$$

(Le Lionnais 1983, p. 35). Boyd (1977) found the following table of small Salem numbers, and suggested that $\sigma_1$, $\sigma_2$, $\sigma_3$, and $\sigma_4$ are the smallest Salem numbers. The Notation 1 1 0 $-1$ $-1$ $-1$ is short for 1 1 0 $-1$ $-1$ $-1$ $-1$ $-1$ 0 1 1, the coefficients of the above polynomial.

$k$	$\sigma_k$	°	Polynomial
1	1.1762808183	10	1 1 0 $-1$ $-1$ $-1$
2	1.1883681475	18	1 $-1$ 1 $-1$ 0 0 $-1$ 1 $-1$ 1
3	1.2000265240	14	1 0 0 $-1$ $-1$ 0 0 1
4	1.2026167437	14	1 0 $-1$ 0 0 0 0 $-1$
5	1.2163916611	10	1 0 0 0 $-1$ $-1$
6	1.2197208590	18	1 $-1$ 0 0 0 0 0 0 $-1$ 1
7	1.2303914344	10	1 0 0 $-1$ 0 $-1$
8	1.2326135486	20	1 $-1$ 0 0 0 $-1$ 1 0 0 $-1$ 1
9	1.2356645804	22	1 0 $-1$ $-1$ 0 0 0 1 1 0 $-1$ $-1$
10	1.2363179318	16	1 $-1$ 0 0 0 0 0 0 $-1$
11	1.2375048212	26	1 0 $-1$ 0 0 $-1$ 0 0 $-1$ 0 1 0 0 1
12	1.2407264237	12	1 $-1$ 1 $-1$ 0 0 $-1$
13	1.2527759374	18	1 0 0 0 0 0 $-1$ $-1$ $-1$ $-1$
14	1.2533306502	20	1 0 $-1$ 0 0 $-1$ 0 0 0 0 0
15	1.2550935168	14	1 0 $-1$ $-1$ 0 1 0 $-1$
16	1.2562211544	18	1 $-1$ 0 0 $-1$ 1 0 0 0 $-1$
17	1.2601035404	24	1 $-1$ 0 0 $-1$ 1 0 $-1$ 1 $-1$ 0 1 $-1$
18	1.2602842369	22	1 $-1$ 0 $-1$ 1 0 0 0 $-1$ 1 $-1$ 1
19	1.2612309611	10	1 0 $-1$ 0 0 $-1$
20	1.2630381399	26	1 $-1$ 0 0 0 0 $-1$ 0 0 0 0 0 0 1
21	1.2672964425	14	1 $-1$ 0 0 0 0 $-1$ 1
22	1.2806381563	8	1 0 0 $-1$ $-1$
23	1.2816913715	26	1 0 0 0 0 0 $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$
24	1.2824955606	20	1 $-2$ 2 $-2$ 2 $-2$ 1 0 $-1$ 1 $-1$
25	1.2846165509	18	1 0 0 0 $-1$ 0 $-1$ $-1$ 0 $-1$
26	1.2847468215	26	1 $-2$ 1 1 $-2$ 1 0 0 $-1$ 1 0 $-1$ 1 $-1$
27	1.2850993637	30	1 0 0 0 0 $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ 0 0 0 0 1
28	1.2851215202	30	1 $-2$ 2 $-2$ 1 0 $-1$ 2 $-2$ 1 0 $-1$ 1 $-1$ 1 $-1$
29	1.2851856708	30	1 $-1$ 0 0 0 0 0 0 $-1$ 0 0 0 $-1$ 0 0 $-1$
30	1.2851967268	26	1 0 $-1$ $-1$ 0 0 0 1 0 $-1$ $-1$ 0 1 1
31	1.2851991792	44	1 $-1$ 0 0 0 0 0 $-1$ 0 0 0 $-1$ 0 0 0 0 0 0 0 1 0 0 1
32	1.2852354362	30	1 0 $-1$ 0 0 $-1$ $-1$ 0 0 0 1 0 0 1 0 $-1$
33	1.2854090648	34	1 $-1$ 0 0 $-1$ 1 $-1$ 0 1 $-1$ 1 0 $-1$ 1 $-1$ 0 1 $-1$
34	1.2863959668	18	1 $-2$ 2 $-2$ 2 $-2$ 2 $-3$ 3 $-3$
35	1.2867301820	26	1 $-1$ 0 0 $-1$ 1 $-1$ 0 1 $-1$ 1 0 $-1$ 1
36	1.2917414257	24	1 $-1$ 0 0 0 0 $-1$ 0 0 0 0 0 0
37	1.2920391602	20	1 0 $-1$ 0 0 $-1$ 0 0 $-1$ 0 1
38	1.2934859531	10	1 0 $-1$ $-1$ 0 1
39	1.2956753719	18	1 $-1$ 0 0 $-1$ 1 $-1$ 0 1 $-1$

See also Pisot-Vijayaraghavan Constants

References

Boyd, D. W. ``Small Salem Numbers.'' Duke Math. J. 44, 315-328, 1977.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.

Lehmer, D. H. ``Factorization of Certain Cyclotomic Functions.'' Ann. Math., Ser. 2 34, 461-479, 1933.

Salem, R. ``Power Series with Integral Coefficients.'' Duke Math. J. 12, 153-172, 1945.

Stewart, C. L. ``Algebraic Integers whose Conjugates Lie Near the Unit Circle.'' Bull. Soc. Math. France 106, 169-176, 1978.