

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
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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 GelfondSchneider 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 GelfondSchneider 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



