Salem Constants

A Salem constant, also called a Salem number, is a real algebraic integer whose conjugates other than itself have absolute value at most 1, with at least one on the unit circle. Its conjugates consist of and numbers on the unit circle (Le Lionnais 1983, p. 150). Every Pisot number is a limit point of Salem constants from both sides (Salem 1945). The smallest known Salem number was found by Lehmer (1933) as the largest real root of

(1)

which is

(2)

(OEIS A073011; Le Lionnais 1983, p. 35). This is the famous constant appearing in Lehmer's Mahler measure problem.

Writing , a cyclotomic polynomial identity is

(3)

where is the th cyclotomic polynomial (Bailey and Broadhurst 1999, p. 6). It can be proved exactly by polynomial reduction, since the difference of the two sides has zero polynomial remainder modulo Lehmer's degree-10 polynomial above. An equivalent identity involving factors is given by Trott (2006).

Boyd (1977) found the following table of small Salem numbers, and suggested that , , , and are the smallest Salem numbers. The notation 1 1 0 is short for 1 1 0 0 1 1, the coefficients of the above polynomial.

			polynomial
1	1.1762808183	10	1 1 0
2	1.1883681475	18	1 1 0 0 1 1
3	1.2000265240	14	1 0 0 0 0 1
4	1.2026167437	14	1 0 0 0 0 0
5	1.2163916611	10	1 0 0 0
6	1.2197208590	18	1 0 0 0 0 0 0 1
7	1.2303914344	10	1 0 0 0
8	1.2326135486	20	1 0 0 0 1 0 0 1
9	1.2356645804	22	1 0 0 0 0 1 1 0
10	1.2363179318	16	1 0 0 0 0 0 0
11	1.2375048212	26	1 0 0 0 0 0 0 1 0 0 1
12	1.2407264237	12	1 1 0 0
13	1.2527759374	18	1 0 0 0 0 0
14	1.2533306502	20	1 0 0 0 0 0 0 0 0
15	1.2550935168	14	1 0 0 1 0
16	1.2562211544	18	1 0 0 1 0 0 0
17	1.2601035404	24	1 0 0 1 0 1 0 1
18	1.2602842369	22	1 0 1 0 0 0 1 1
19	1.2612309611	10	1 0 0 0
20	1.2630381399	26	1 0 0 0 0 0 0 0 0 0 0 1
21	1.2672964425	14	1 0 0 0 0 1
22	1.2806381563	8	1 0 0
23	1.2816913715	26	1 0 0 0 0 0
24	1.2824955606	20	1 2 2 1 0 1
25	1.2846165509	18	1 0 0 0 0 0
26	1.2847468215	26	1 1 1 1 0 0 1 0 1
27	1.2850993637	30	1 0 0 0 0 0 0 0 0 1
28	1.2851215202	30	1 2 1 0 2 1 0 1 1
29	1.2851856708	30	1 0 0 0 0 0 0 0 0 0 0 0
30	1.2851967268	26	1 0 0 0 0 1 0 0 1 1
31	1.2851991792	44	1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1
32	1.2852354362	30	1 0 0 0 0 0 0 1 0 0 1 0
33	1.2854090648	34	1 0 0 1 0 1 1 0 1 0 1
34	1.2863959668	18	1 2 2 2 3
35	1.2867301820	26	1 0 0 1 0 1 1 0 1
36	1.2917414257	24	1 0 0 0 0 0 0 0 0 0 0
37	1.2920391060	20	1 0 0 0 0 0 0 1
38	1.2934859531	10	1 0 0 1
39	1.2956753719	18	1 0 0 1 0 1

Cheng et al. (2026) proved that the unbiased Bernoulli convolution with parameter has Hausdorff dimension strictly less than 1 for every quartic Salem constant . Cheng et al. (2026) report AI assistance with mathematical checks.