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# Salem Constants

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

Lehmer's Mahler Measure Problem, Pisot Number

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## References

Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44, 315-328, 1977.Boyd, D. W. "Pisot and Salem Numbers in Intervals of the Real Line." Math. Comput. 32, 1244-1260, 1978.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.Mossinghoff, M. "Small Salem Numbers." http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html.Salem, R. "Power Series with Integral Coefficients." Duke Math. J. 12, 153-172, 1945.Sloane, N. J. A. Sequence A073011 in "The On-Line Encyclopedia of Integer Sequences."Stewart, C. L. "Algebraic Integers whose Conjugates Lie Near the Unit Circle." Bull. Soc. Math. France 106, 169-176, 1978.

Salem Constants

## Cite this as:

Weisstein, Eric W. "Salem Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SalemConstants.html