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

%I A073011 #50 Feb 16 2025 08:32:46
%S A073011 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,
%T A073011 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,
%U A073011 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
%N A073011 Decimal expansion of Lehmer's constant (also known as the Salem constant).
%C A073011 This number is algebraic of degree 10.
%C A073011 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
%C A073011 The Salem numbers were named after the Greek mathematician Raphaël Salem (1898-1963). - _Amiram Eldar_, May 01 2021
%D A073011 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30, p. 193.
%H A073011 G. C. Greubel, <a href="/A073011/b073011.txt">Table of n, a(n) for n = 1..5000</a>
%H A073011 David Boyd, <a href="https://dx.doi.org/10.1215/S0012-7094-77-04413-1">Small Salem numbers</a>, Duke Math. Journal, vol. 44, 1977, pp. 315-328.
%H A073011 Henri Cohen, Leonard Lewin, and Don Zagier. <a href="http://projecteuclid.org/euclid.em/1048709113">A sixteenth-order polylogarithm ladder</a>, Experimental Mathematics 1.1 (1992): 25-34.
%H A073011 Eriko Hironaka, <a href="http://www.math.fsu.edu/~aluffi/archive/paper355.pdf">What is Lehmer's number?</a>, Notices Amer. Math. Soc., 56 (No. 3, 2009), 374-375.
%H A073011 D. H. Lehmer, <a href="http://www.jstor.org/stable/1968172">Factorization of certain cyclotomic functions</a>, Annals of Math. vol. 34, 1933, pp. 461-479.
%H A073011 Douglas Lind, <a href="https://arxiv.org/abs/math/0303279">Lehmer's Problem for compact abelian groups</a>, arXiv:math/0303279 [math.NT], 2003-2014.
%H A073011 Michael Mossinghoff, <a href="https://web.archive.org/web/20131027202648/http://oldweb.cecm.sfu.ca/~mjm/Lehmer/">Lehmer's Problem Website</a>.
%H A073011 Michael Mossinghoff, <a href="https://web.archive.org/web/20210223153647/http://www.cecm.sfu.ca/~mjm/Lehmer/lists/SalemList.html">Small Salem Numbers</a>.
%H A073011 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap81.html">Salem Constant</a>.
%H A073011 Raphaël Salem, <a href="http://doi.org/10.1215/S0012-7094-45-01213-0">Power series with integral coefficients</a>, Duke mathematical journal, Vol. 12, No. 1 (1945), pp. 153-172.
%H A073011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SalemConstants.html">Salem Constants</a>.
%H A073011 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H A073011 <a href="/index/Al#algebraic_10">Index entries for algebraic numbers, degree 10</a>
%F A073011 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.
%e A073011 1.17628081825991750654407033847403505069341580656469...
%t A073011 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 *)
%t A073011 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 *)
%o A073011 (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))
%o A073011 (PARI) polrootsreal(Pol([1, 1, 0, -1, -1, -1, -1, -1, 0, 1, 1]))[2] \\ _Charles R Greathouse IV_, Sep 03 2014
%Y A073011 Cf. A070178 (Coefficients of Lehmer's polynomial).
%K A073011 cons,nonn
%O A073011 1,3
%A A073011 _Robert G. Wilson v_, Aug 03 2002
%E A073011 Edited by _N. J. A. Sloane_, May 01 2012