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

FORMULA

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

EXAMPLE

0.76666469596212309311120442251031484800...

MAPLE

evalf((f-> LambertW(f)/f)(log(2)/2), 145); # Alois P. Heinz, Aug 03 2023

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 *)

RealDigits[-x/.FindRoot[2^x==x^2, {x, -1}, WorkingPrecision->120], 10, 120][[1]] (* Harvey P. Dale, Jul 15 2023 *)

PROG

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

CROSSREFS

KEYWORD

nonn,cons

AUTHOR

Robert G. Wilson v, Aug 17 2002

EXTENSIONS

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

STATUS

approved