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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 >1 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

 x^(10)+x^9-x^7-x^6-x^5-x^4-x^3+x+1=0,

which is

 sigma_1=1.176280818...

(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 sigma_1, sigma_2, sigma_3, and sigma_4 are the smallest Salem numbers. The notation 1 1 0 -1 -1 -1 is short for 1 1 0 -1 -1 -1 -1 -1 0 1 1, the coefficients of the above polynomial.

ksigma_k degreespolynomial
11.1762808183101 1 0 -1 -1 -1
21.1883681475181 -1 1 -1 0 0 -1 1 -1 1
31.2000265240141 0 0 -1 -1 0 0 1
41.2026167437141 0 -1 0 0 0 0 -1
51.2163916611101 0 0 0 -1 -1
61.2197208590181 -1 0 0 0 0 0 0 -1 1
71.2303914344101 0 0 -1 0 -1
81.2326135486201 -1 0 0 0 -1 1 0 0 -1 1
91.2356645804221 0 -1 -1 0 0 0 1 1 0 -1 -1
101.2363179318161 -1 0 0 0 0 0 0 -1
111.2375048212261 0 -1 0 0 -1 0 0 -1 0 1 0 0 1
121.2407264237121 -1 1 -1 0 0 -1
131.2527759374181 0 0 0 0 0 -1 -1 -1 -1
141.2533306502201 0 -1 0 0 -1 0 0 0 0 0
151.2550935168141 0 -1 -1 0 1 0 -1
161.2562211544181 -1 0 0 -1 1 0 0 0 -1
171.2601035404241 -1 0 0 -1 1 0 -1 1 -1 0 1 -1
181.2602842369221 -1 0 -1 1 0 0 0 -1 1 -1 1
191.2612309611101 0 -1 0 0 -1
201.2630381399261 -1 0 0 0 0 -1 0 0 0 0 0 0 1
211.2672964425141 -1 0 0 0 0 -1 1
221.280638156381 0 0 -1 -1
231.2816913715261 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1
241.2824955606201 -2 2 -2 2 -2 1 0 -1 1 -1
251.2846165509181 0 0 0 -1 0 -1 -1 0 -1
261.2847468215261 -2 1 1 -2 1 0 0 -1 1 0 -1 1 -1
271.2850993637301 0 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 0 1
281.2851215202301 -2 2 -2 1 0 -1 2 -2 1 0 -1 1 -1 1 -1
291.2851856708301 -1 0 0 0 0 0 0 -1 0 0 0 -1 0 0 -1
301.2851967268261 0 -1 -1 0 0 0 1 0 -1 -1 0 1 1
311.2851991792441 -1 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 1
321.2852354362301 0 -1 0 0 -1 -1 0 0 0 1 0 0 1 0 -1
331.2854090648341 -1 0 0 -1 1 -1 0 1 -1 1 0 -1 1 -1 0 1 -1
341.2863959668181 -2 2 -2 2 -2 2 -3 3 -3
351.2867301820261 -1 0 0 -1 1 -1 0 1 -1 1 0 -1 1
361.2917414257241 -1 0 0 0 0 -1 0 0 0 0 0 0
371.2920391602201 0 -1 0 0 -1 0 0 -1 0 1
381.2934859531101 0 -1 -1 0 1
391.2956753719181 -1 0 0 -1 1 -1 0 1 -1

See also

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.

Referenced on Wolfram|Alpha

Salem Constants

Cite this as:

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

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