|
%I #25 Oct 27 2023 09:39:28
%S 7,8,4,1,9,0,3,7,3,3,7,7,1,2,2,2,4,7,1,0,8,3,9,5,4,7,7,8,1,5,6,8,7,7,
%T 5,2,6,5,3,8,6,7,4,9,4,4,5,1,3,5,9,2,0,6,4,5,3,5,7,5,5,3,9,7,5,5,6,8,
%U 6,7,8,7,3,9,5,6,6,8,3,7,3,9,0,0,3,8,3,1,4,4,6,7,4,6,2,8,9,3,3,7,6,4,1,9,4,3,0,0,2,5,8,4,7,1,7,5,7,2,1
%N Decimal expansion of (3*sqrt(3)+sqrt(7))/10.
%C Smith & Smith conjecture that this is the Steiner ratio rho_3, the least upper bound on the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 3. Diaconis & Graham offer $1000 for proof (or disproof) of this conjecture.
%C This is an algebraic number of degree 4; the minimal polynomial is 25x^4 - 17x^2 + 1.
%D Persi Diaconis and R. L. Graham, Magical Mathematics: The Mathematical Ideas that Animate Great Magic Tricks, Princeton University Press, 2011. See pp. 212-214.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.6 Steiner Tree Constants, p. 504.
%H Warren D. Smith and J. MacGregor Smith, <a href="http://dx.doi.org/10.1016/0097-3165(95)90055-1">On the Steiner ratio in 3-space</a>, Journal of Combinatorial Theory, Series A 69:2 (1995), pp. 301-332.
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F (3*sqrt(3)+sqrt(7))/10.
%e 0.7841903733771222471083954778156877526538674944513592064535755...
%t RealDigits[(3*Sqrt[3]+Sqrt[7])/10, 10, 120] // First (* _Jean-François Alcover_, May 27 2014 *)
%o (PARI) (3*sqrt(3)+sqrt(7))/10
%o (PARI) polrootsreal(25*x^4 - 17*x^2 + 1)[4] \\ _Charles R Greathouse IV_, Jan 05 2016
%Y Cf. A010527.
%K nonn,cons
%O 0,1
%A _Charles R Greathouse IV_, Dec 11 2012
%E Formula and name simplified by _Jean-François Alcover_, May 27 2014
|
