释义 |
Salem ConstantsEach point of the Pisot-Vijayaraghavan Constants is a Limit Point from both sides of a set known as theSalem constants (Salem 1945). The Salem constants are algebraic Integers in which one or more of theconjugates is on the Unit Circle with the others inside (Le Lionnais 1983, p. 150). The smallest known Salem numberwas found by Lehmer (1933) as the largest Real Root of
 which is
 (Le Lionnais 1983, p. 35). 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 for1 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  |
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.
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