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A073084 Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2. 5
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

From Robert G. Wilson v, May 18 2021: (Start)

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)

REFERENCES

"Angela" (R. J. Milazzo, rgmilazzo(AT)aol.com), Posting to the sci.math usenet Aug 17, 2002.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..1999

Eric Weisstein's World of Mathematics, Power

Index entries for transcendental numbers

FORMULA

-2*LambertW(log(2)/2)/log(2). - Eric W. Weisstein, Jan 23 2005

EXAMPLE

0.76666469596212309311120442251031484800...

MATHEMATICA

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 *)

PROG

(PARI) lambertw(log(sqrt(2))/log(sqrt(2)) \\ Stanislav Sykora, Nov 04 2013

CROSSREFS

Cf. A010503 (decimal expansion of sqrt(2)/2).

Sequence in context: A199871 A103616 A290570 * A011473 A259171 A021570

Adjacent sequences:  A073081 A073082 A073083 * A073085 A073086 A073087

KEYWORD

nonn,cons

AUTHOR

Robert G. Wilson v, Aug 17 2002

EXTENSIONS

Offset corrected by R. J. Mathar, Feb 05 2009

STATUS

approved

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Last modified October 22 01:48 EDT 2021. Contains 348160 sequences. (Running on oeis4.)