A073084 Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2.

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

Offset: 0

Author

Robert G. Wilson v, Aug 17 2002

Keywords

• nonn
• cons

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)

x is also the negative solution to 4^x = x^4, which reduces to 2^x = x^2 upon taking the square root of both sides. - Jason Bard, Aug 16 2025

Examples

			0.76666469596212309311120442251031484800...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..1999
• RJMilazzo and others, largest solution to 2^x=x^2, thread in newsgroup sci.math, Aug 17, 2002.
• Eric Weisstein's World of Mathematics, Power.
• Index entries for transcendental numbers.

Crossrefs

Cf. A010503, A344905.

Cf. non-integer solutions to a^x = x^a: A166928 (a = 3), A341328 (a = 5).

Programs

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 *)

PARI

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

Formula

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

Equals 1/A344905. - Hugo Pfoertner, Dec 18 2024

Extensions

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