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A073084
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Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2.
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7
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7, 6, 6, 6, 6, 4, 6, 9, 5, 9, 6, 2, 1, 2, 3, 0, 9, 3, 1, 1, 1, 2, 0, 4, 4, 2, 2, 5, 1, 0, 3, 1, 4, 8, 4, 8, 0, 0, 6, 6, 7, 5, 3, 4, 6, 6, 6, 9, 8, 3, 2, 0, 5, 8, 4, 6, 0, 8, 8, 4, 3, 7, 6, 9, 3, 5, 5, 5, 2, 7, 9, 5, 7, 2, 4, 8, 7, 2, 4, 2, 2, 8, 5, 3, 0, 2, 9, 2, 0, 9, 6, 9, 7, 9, 0, 2, 5, 3, 0, 5, 6, 5, 4, 7, 9
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OFFSET
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0,1
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COMMENTS
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The equation has three solutions, x = 2, 4 and -0.76666469596....
-x is the power tower (tetration) of 1/sqrt(2) (A010503), also equal to LambertW(log(sqrt(2))/log(sqrt(2)). - Stanislav Sykora, Nov 04 2013
x is transcendental by the Gelfond-Schneider theorem. Proof: If we accept that x is not an integer, then we can see that x is not rational. For if it were, x^2 would be as well, whereas 2^x would not be (because 2 is not a perfect power). Thus we would have a contradiction (since x^2 = 2^x). Similarly, if x were irrational algebraic, x^2 would be as well, while 2^x would be transcendental (by the Gelfond-Schneider theorem). Thus the only conclusion is that x is transcendental. - Chayim Lowen, Aug 13 2015
Let W be the Lambert power log function,
f(x) = e^(-W_x(-log(sqrt(2)))) and g(x) = -e^(-W_x(log(sqrt(2)))).
Then f(0)=2, f(-1)= 4 and g(0) = c. Except for these three illustrated examples, all integer arguments x yield a complex solution which satisfies the equation.
(End)
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REFERENCES
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"Angela" (R. J. Milazzo, rgmilazzo(AT)aol.com), Posting to the sci.math usenet Aug 17, 2002.
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LINKS
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Eric Weisstein's World of Mathematics, Power
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FORMULA
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EXAMPLE
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0.76666469596212309311120442251031484800...
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MAPLE
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evalf((f-> LambertW(f)/f)(log(2)/2), 145); # Alois P. Heinz, Aug 03 2023
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MATHEMATICA
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RealDigits[NSolve[2^x == x^2, x, WorkingPrecision -> 150][[1, 1]][[2]]][[1]]
c = -Exp[-LambertW[Log[2]/2]]; RealDigits[c, 10, 111][[1]] (* Robert G. Wilson v, May 18 2021 *)
(* To view the two curves: *) Plot[{2^x, x^2}, {x, -4.5, 4.5}] (* Robert G. Wilson v, May 18 2021 *)
RealDigits[-x/.FindRoot[2^x==x^2, {x, -1}, WorkingPrecision->120], 10, 120][[1]] (* Harvey P. Dale, Jul 15 2023 *)
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PROG
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CROSSREFS
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Cf. A010503 (decimal expansion of sqrt(2)/2).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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