A372270 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.

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

Offset: 0

Author

Pontus von Brömssen, Apr 25 2024

Keywords

• nonn
• cons

Examples

			0.906179845938663992797626878299392965125651910762530862873762...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Wikipedia, Legendre polynomials.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A008316, A100258.

There are floor(k/2) positive zeros of the Legendre polynomial of degree k:

k | zeros

---+--------------------------

2 | A020760

3 | A010513/10

4 | A372267, A372268

5 | A372269, A372270

6 | A372271, A372272, A372273

7 | A372274, A372275, A372276

Programs

Mathematica

First[RealDigits[Root[LegendreP[5, #] &, 5], 10, 100]] (* Paolo Xausa, Feb 27 2025 *)

Formula

Largest positive root of 63*x^4 - 70*x^2 + 15 = 0.

Equals sqrt(5+2*sqrt(10/7))/3.