A160155 Decimal expansion of the one real root of x^5-x-1.

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

Offset: 1

Author

Harry J. Smith, May 03 2009

Keywords

• nonn
• easy
• cons

Comments

The other (complex) roots are 0.181232444469875383... + 1.08395410131771066...*i, and -0.764884433600584726... + 0.352471546031726249...*i, together with their complex conjugates. - Wolfdieter Lang, Dec 15 2022

This quintic is in some sense the smallest and/or simplest algebraic equation for which there is no explicit expression for the roots. (The "equivalent" quintic x^5 - x + 1 has the opposite real root, x = -1.1673..., while x^5 + x + 1 = (x^2 + x + 1)(x^3 - x^2 + 1).) - M. F. Hasler, Jul 12 2025

Examples

			1.16730397826141868425604589985484218072056037152548903914008244927565...
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• David W. Boys, The maximal modulus of an algebraic integer, Math. Comp. 45 (1985) 243-249, table page S18.
• Wikipedia, Abel-Ruffini theorem, created Nov. 27, 2002, retrieved July 12, 2025.
• Qiang Wu, The smallest Perron numbers, Math. Comp. 79 (2010) 2387-2394.
• Index entries for algebraic numbers, degree 5

Crossrefs

Cf. A039922 (continued fraction), A001622 (golden ratio phi = root of x^2 - x - 1), A060006 (plastic constant, root of x^3 - x - 1), A060007 (root of x^4 - x - 1).

Programs

Mathematica

RealDigits[Root[x^5-x-1, x, 1], 10, 105] // First (* Jean-François Alcover, Jul 09 2015 *)

PARI

localprec(20080); r=real(polroots('x^5 - 'x - 1)[1]); for (n=1, 20000, d=floor(r); r=(r-d)*10; write("b160155.txt", n, " ", d)) \\ Edited by M. F. Hasler, Jul 12 2025

PARI

polrootsreal(x^5-x-1)[1] \\ Charles R Greathouse IV, Apr 14 2014

Formula

Equals (1 + (1 + (1 + (1 + (1 + ...)^(1/5))^(1/5))^(1/5))^(1/5))^(1/5). - Ilya Gutkovskiy, Dec 15 2017