Definition:Salem Constant
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Theorem
The Salem constant is the greatest real root of Lehmer's polynomial:
- $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 = 0$
Its value is approximately:
- $1 \cdotp 17628 \, 08182 \, 599 \ldots$
This sequence is A073011 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also known as
The Salem constant is also known as Lehmer's constant, for Derrick Henry Lehmer, but there is already a constant with that name.
Source of Name
This entry was named for Raphaël Salem.
Historical Note
The Salem constant was discovered by Derrick Henry Lehmer in $1933$.
It was conjectured in $1977$ by David William Boyd that it is the smallest of the Salem numbers.
Sources
- 1933: D.H. Lehmer: Factorisation of certain cyclotomic functions (Ann. Math. Ser. 2 Vol. 34: pp. 461 – 479) www.jstor.org/stable/1968172
- 1977: David W. Boyd: Small Salem Numbers (Duke Math. J. Vol. 44: pp. 315 – 328)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,17628 081 \ldots$
- Weisstein, Eric W. "Salem Constants." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SalemConstants.html