## Salem Constants

Each point of the Pisot-Vijayaraghavan Constants is a Limit Point from both sides of a set known as the Salem constants (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

(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 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

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.