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A348359
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Decimal expansion of the nontrivial number x for which x^phi = phi^x, where phi is the golden ratio (1+sqrt(5))/2.
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0
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6, 0, 5, 5, 7, 2, 2, 0, 9, 1, 0, 2, 4, 7, 4, 1, 0, 0, 2, 1, 2, 6, 6, 3, 9, 1, 1, 7, 5, 8, 3, 1, 4, 9, 7, 3, 1, 6, 8, 3, 8, 2, 8, 7, 5, 3, 7, 8, 3, 6, 7, 7, 7, 4, 3, 9, 4, 9, 9, 6, 7, 7, 3, 5, 2, 8, 1, 8, 7, 9, 7, 4, 4, 8, 5, 2, 3, 5, 8, 1, 4, 7, 9, 3, 8, 9, 4, 6, 6, 6, 0, 7, 4, 2, 8, 1, 7, 8, 9, 4, 7, 8, 9, 4, 5, 7
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OFFSET
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1,1
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COMMENTS
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The x-th root of x equals the phi-th root of phi: x^(1/x) = phi^(1/phi) = A185261 = 1.3463608200348694434247534661858... .
Not surprisingly, x appears to be irrational. If x is also algebraic, then x^phi would be transcendental by the Gelfond-Schneider theorem.
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LINKS
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EXAMPLE
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6.055722091024741002126639117583149731683828...
x^phi = phi^x = 18.431940924839652158136364051482054378959672... .
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MATHEMATICA
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{a, b} = NSolve[x^phi == phi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[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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