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A320157 Decimal expansion of real root of x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296 = 0, x^6*(x + 7) - 6^6 = 0. 3

%I #26 Aug 21 2023 12:00:32

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

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

%U 8,8,7,6,1,3,8,3,9,6,7,0,5,9,8,1,6,5,0

%N Decimal expansion of real root of x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296 = 0, x^6*(x + 7) - 6^6 = 0.

%C Let t_1, t_2, ..., t_n be n real numbers in [-1, 1] such that Sum_{i=1..n} (t_i)^k = 0, then lim sup ((Sum_{i=1..n} (t_i)^m)/n) = (x_0)/k, where k > m > 0 are odd numbers and x_0 is the unique positive root of x^(k-m)*(x + k)^m - m^m*(k - m)^(k-m). x_0 is an algebraic integer of degree k - 2. The upper bound (x_0)/k can be approached by x_1 = x_2 = ... = x_(p^k) = -1, x_(p^k+1) = x_(p^k+2) = ... = x_(p^k+q^k) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 is the unique positive root of (k - m)*x^k + k*x^(k - m) - m = 0. Note that (x_0)/k = ((t_0)^m - (t_0)^k)/((t_0)^k + 1).

%C Here k = 7, m = 1, so the upper bound is (x_0)/7 = 0.57457032651694013806558..., where x_0 = 4.0219922856185809664591... is the unique positive root to x^6*(x + 7) - 6^6 = 0. The upper bound (x_0)/5 can be approached by t_1 = t_2 = ... = t_(p^7) = -1, t_(p^7+1) = t_(p^7+2) = ... = t_(p^7+q^7) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 = 0.67033204760309682774318... is the unique positive root of 6x^7 + 7x^6 - 1 = 0. For example, let p = 67033, q = 100000, t_1 = t_2 = ... = t_(67033^7) = -1, t_(67033^7+1) = t_(67033^7+2) = ... = t_(67033^7+100000^7) = 0.67033, then (Sum_{i=1..67033^7+100000^7} t_i)/(67033^7 + 100000^7) = 0.57457032649925207012850..., very close to (x_0)/7. Note that (x_0)/7 = (t_0 - (t_0)^7)/((t_0)^7 + 1).

%H Jianing Song, <a href="/A320157/b320157.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Al#algebraic_05">Index entries for algebraic numbers, degree 5</a>

%e 4.0219922856185809664591185627085393007453409109840...

%p evalf(solve(x^5-5*x^4+24*x^3-108*x^2+432*x-1296=0,x)[1],120); # _Muniru A Asiru_, Oct 07 2018

%t Part[RealDigits[N[Root[x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296,1], 100]], 1] (* _Stefano Spezia_, Oct 07 2018 *)

%o (PARI) default(realprecision, 100); solve(x=4, 5, x^5 - 5*x^4 + 24*x^3 - 108*x^2 + 432*x - 1296)

%Y Similar sequences: A320156 (k=5, m=1), A320158 (k=5, m=3).

%K nonn,cons

%O 1,1

%A _Jianing Song_, Oct 06 2018

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)