Salem constants, sometimes also called Salem numbers, are a set of numbers of which each point of a Pisot number is a limit point from both sides (Salem 1945). The Salem constants are algebraic integers 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
which is
(OEIS A073011; Le Lionnais 1983, p. 35). This is the famous constant appearing in Lehmer's Mahler measure problem.
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 | 1.2920391602 | 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 |