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

%I A073084 #65 Aug 16 2025 10:04:20
%S A073084 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,
%T A073084 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,
%U A073084 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
%N A073084 Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2.
%C A073084 The equation has three solutions, x = 2, 4 and -0.76666469596....
%C A073084 -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
%C A073084 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
%C A073084 From _Robert G. Wilson v_, May 18 2021: (Start)
%C A073084 Let W be the Lambert power log function,
%C A073084 f(x) = e^(-W_x(-log(sqrt(2)))) and g(x) = -e^(-W_x(log(sqrt(2)))).
%C A073084 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)
%C A073084 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
%D A073084 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.
%H A073084 Stanislav Sykora, <a href="/A073084/b073084.txt">Table of n, a(n) for n = 0..1999</a>
%H A073084 RJMilazzo and others, <a href="https://groups.google.com/g/sci.math/c/_xDF0WhaOdE/m/PQi3-OURjgEJ">largest solution to 2^x=x^2</a>, thread in newsgroup sci.math, Aug 17, 2002.
%H A073084 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Power.html">Power</a>.
%H A073084 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A073084 -2*LambertW(log(2)/2)/log(2). - _Eric W. Weisstein_, Jan 23 2005
%F A073084 Equals 1/A344905. - _Hugo Pfoertner_, Dec 18 2024
%e A073084 0.76666469596212309311120442251031484800...
%p A073084 evalf((f-> LambertW(f)/f)(log(2)/2), 145);  # _Alois P. Heinz_, Aug 03 2023
%t A073084 RealDigits[NSolve[2^x == x^2, x, WorkingPrecision -> 150][[1, 1]][[2]]][[1]]
%t A073084 c = -Exp[-LambertW[Log[2]/2]]; RealDigits[c, 10, 111][[1]] (* _Robert G. Wilson v_, May 18 2021 *)
%t A073084 (* To view the two curves: *) Plot[{2^x, x^2}, {x, -4.5, 4.5}] (* _Robert G. Wilson v_, May 18 2021 *)
%t A073084 RealDigits[-x/.FindRoot[2^x==x^2,{x,-1},WorkingPrecision->120],10,120][[1]] (* _Harvey P. Dale_, Jul 15 2023 *)
%o A073084 (PARI) lambertw(log(sqrt(2)))/log(sqrt(2)) \\ _Stanislav Sykora_, Nov 04 2013
%Y A073084 Cf. A010503, A344905.
%Y A073084 Cf. non-integer solutions to a^x = x^a: A166928 (a = 3), A341328 (a = 5).
%K A073084 nonn,cons
%O A073084 0,1
%A A073084 _Robert G. Wilson v_, Aug 17 2002
%E A073084 Offset corrected by _R. J. Mathar_, Feb 05 2009