A124930 Decimal expansion of the unique positive real root of the equation x^x = x + 1.

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, 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, 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

Offset: 1

Author

Rick L. Shepherd, Nov 12 2006

Keywords

• cons
• nonn

Comments

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

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

Examples

			1.77677504009705469747973074403...
		

Links

• R. P. Stanley, A transcendental number?: Quickie 88-10, Mathematical Entertainments column (Steven H. Weintraub editor), The Mathematical Intelligencer, Vol. 11, No. 1, Winter 1989, p. 55.
• Index entries for transcendental numbers

Programs

Mathematica

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

PARI

solve(x=1, 2, x^x-x-1)