A073011 Decimal expansion of Lehmer's constant (also known as the Salem constant).

1, 1, 7, 6, 2, 8, 0, 8, 1, 8, 2, 5, 9, 9, 1, 7, 5, 0, 6, 5, 4, 4, 0, 7, 0, 3, 3, 8, 4, 7, 4, 0, 3, 5, 0, 5, 0, 6, 9, 3, 4, 1, 5, 8, 0, 6, 5, 6, 4, 6, 9, 5, 2, 5, 9, 8, 3, 0, 1, 0, 6, 3, 4, 7, 0, 2, 9, 6, 8, 8, 3, 7, 6, 5, 4, 8, 5, 4, 9, 9, 6, 2, 0, 9, 6, 8, 3, 0, 1, 1, 5, 5, 8, 1, 8, 1, 5, 3, 9, 4, 6, 5, 9, 2, 0

Offset: 1

Author

Robert G. Wilson v, Aug 03 2002

Keywords

• cons
• nonn

Comments

This number is algebraic of degree 10.

The Salem constant given here is the smallest known value of Mahler's measure M(f)=abs(a_d)*Product_{k=1..d}max(1,abs(b_k)) of a polynomial f(x)=Sum_{k=0..d}(a_k*x^k)=a_d*Product_{k=1..d}(x-b_k). The minimum occurs for Lehmer's polynomial (coefficients A070178) L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1 with M(L)=1.1762808... - Hugo Pfoertner, Mar 12 2006

The Salem numbers were named after the Greek mathematician Raphaël Salem (1898-1963). - Amiram Eldar, May 01 2021

Examples

			1.17628081825991750654407033847403505069341580656469...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30, p. 193.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• David Boyd, Small Salem numbers, Duke Math. Journal, vol. 44, 1977, pp. 315-328.
• Henri Cohen, Leonard Lewin, and Don Zagier. A sixteenth-order polylogarithm ladder, Experimental Mathematics 1.1 (1992): 25-34.
• Eriko Hironaka, What is Lehmer's number?, Notices Amer. Math. Soc., 56 (No. 3, 2009), 374-375.
• D. H. Lehmer, Factorization of certain cyclotomic functions, Annals of Math. vol. 34, 1933, pp. 461-479.
• Douglas Lind, Lehmer's Problem for compact abelian groups, arXiv:math/0303279 [math.NT], 2003-2014.
• Michael Mossinghoff, Lehmer's Problem Website.
• Michael Mossinghoff, Small Salem Numbers.
• Simon Plouffe, Salem Constant.
• Raphaël Salem, Power series with integral coefficients, Duke mathematical journal, Vol. 12, No. 1 (1945), pp. 153-172.
• Eric Weisstein's World of Mathematics, Salem Constants.
• Eric Weisstein's World of Mathematics, Polylogarithm.
• Index entries for algebraic numbers, degree 10

Crossrefs

Cf. A070178 (Coefficients of Lehmer's polynomial).

Programs

Mathematica

RealDigits[x/.FindRoot[x^10+x^9-Total[x^Range[3,7]]+x+1==0,{x,1,2}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Sep 08 2011 *)
Root[ x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)

PARI

default(realprecision,250); L(x)=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1; solve(x=1.1,1.2,L(x))

PARI

polrootsreal(Pol([1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1]))[2] \\ Charles R Greathouse IV, Sep 03 2014

Formula

This is the largest real root of the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.

Extensions

Edited by N. J. A. Sloane, May 01 2012