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A320157
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Decimal expansion of real root of x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296 = 0, x^6*(x + 7) - 6^6 = 0.
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3
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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, 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, 8, 8, 7, 6, 1, 3, 8, 3, 9, 6, 7, 0, 5, 9, 8, 1, 6, 5, 0
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OFFSET
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1,1
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COMMENTS
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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).
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).
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LINKS
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EXAMPLE
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4.0219922856185809664591185627085393007453409109840...
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MAPLE
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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
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MATHEMATICA
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Part[RealDigits[N[Root[x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296, 1], 100]], 1] (* Stefano Spezia, Oct 07 2018 *)
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PROG
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(PARI) default(realprecision, 100); solve(x=4, 5, x^5 - 5*x^4 + 24*x^3 - 108*x^2 + 432*x - 1296)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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