A124930 - OEIS

%I #15 Aug 12 2022 19:12:08

%S 1,7,7,6,7,7,5,0,4,0,0,9,7,0,5,4,6,9,7,4,7,9,7,3,0,7,4,4,0,3,8,7,5,6,

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

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

%N Decimal expansion of the unique positive real root of the equation x^x = x + 1.

%C The proof by R. P. Stanley using contradiction and the Gelfond-Schneider Theorem shows that this number is transcendental.

%C Let r be this constant and f(x) be the function x^(1/(r-1)). Since r^(r-1) = 1 + 1/r, we have r = f(1 + 1/f(1 + 1/f(1 + 1/f(1 + ...)))). - _Gerald McGarvey_, Jan 12 2008

%H R. P. Stanley, <a href="https://vdocuments.mx/mathematical-entertainments-57bf89e029909.html?page=3">A transcendental number?: Quickie 88-10</a>, Mathematical Entertainments column (Steven H. Weintraub editor), The Mathematical Intelligencer, Vol. 11, No. 1, Winter 1989, p. 55.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 1.77677504009705469747973074403...

%t RealDigits[x/.FindRoot[x^x==x+1,{x,1.8},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Aug 19 2019 *)

%o (PARI) solve(x=1, 2, x^x-x-1)

%K cons,nonn

%O 1,2

%A _Rick L. Shepherd_, Nov 12 2006