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Decimal expansion of the real positive solution to x^4 = x+4.
1

%I #40 Oct 27 2023 12:07:56

%S 1,5,3,3,7,5,1,1,6,8,7,5,5,2,0,4,2,8,8,1,1,8,0,4,1,4,1,3,2,8,2,2,2,6,

%T 8,3,6,7,0,9,7,2,3,6,3,4,5,5,0,4,8,2,0,8,1,5,4,2,0,3,7,6,0,3,9,6,8,3,

%U 9,9,1,9,9,8,9,1,5,6,1,6,2,4,6,9,3,9,8,8,1,7,6,2,3,3,5,5,9

%N Decimal expansion of the real positive solution to x^4 = x+4.

%C x = (4 + (4 + (4 + ... )^(1/4))^(1/4))^(1/4).

%C The negative value (-1.5337511687...) is the real negative solution to x^4 = 4-x.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F Equals sqrt(sqrt(1/s) - s/16) + sqrt(s/16) where s = (sqrt(16804864/27) + 32)^(1/3) - (sqrt(16804864/27) - 32)^(1/3). [Simplified by _Michal Paulovic_, Jun 22 2021]

%e 1.5337511687552...

%t RealDigits[x /. FindRoot[x^4 - x - 4, {x, 1}, WorkingPrecision -> 100], 10, 90][[1]] (* _Amiram Eldar_, Sep 03 2020 *)

%o (PARI) solve(n=0,2,n^4-n-4)

%o (PARI) polroots(n^4-n-4)[2]

%o (PARI) polrootsreal(n^4-n-4)[2] \\ _Charles R Greathouse IV_, Oct 27 2023

%o (MATLAB) format long; solve('x^4-x-4=0'); ans(1), (eval(ans))

%Y Cf. A337570, A060007, A294644.

%K nonn,cons

%O 1,2

%A _Michal Paulovic_, Sep 01 2020