A073084 - OEIS

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

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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 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

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.

FORMULA

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

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

EXAMPLE

0.76666469596212309311120442251031484800...

MAPLE

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