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A124930
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Decimal expansion of the unique positive real root of the equation x^x = x + 1.
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3
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1, 7, 7, 6, 7, 7, 5, 0, 4, 0, 0, 9, 7, 0, 5, 4, 6, 9, 7, 4, 7, 9, 7, 3, 0, 7, 4, 4, 0, 3, 8, 7, 5, 6, 7, 4, 8, 6, 3, 7, 4, 1, 1, 0, 3, 4, 3, 2, 9, 2, 9, 6, 1, 3, 9, 0, 8, 4, 3, 7, 4, 0, 1, 5, 2, 7, 3, 1, 1, 8, 6, 5, 8, 9, 3, 2, 8, 2, 4, 7, 7, 0, 7, 0, 2, 0, 7, 2, 7, 8, 6, 1, 5, 1, 3, 1, 3, 5, 2, 3, 6, 3, 0, 0, 9
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OFFSET
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1,2
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COMMENTS
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The proof by R. P. Stanley using contradiction and the Gelfond-Schneider Theorem shows that this number is transcendental.
Let r be this constant and f(x) be the function x^(1/(r-1)). Since r^(r-1) = 1 + 1/r, we have r = f(1 + 1/f(1 + 1/f(1 + 1/f(1 + ...)))). - Gerald McGarvey, Jan 12 2008
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LINKS
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EXAMPLE
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1.77677504009705469747973074403...
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MATHEMATICA
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RealDigits[x/.FindRoot[x^x==x+1, {x, 1.8}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 19 2019 *)
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PROG
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(PARI) solve(x=1, 2, x^x-x-1)
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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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