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

%I A372270 #14 Feb 27 2025 04:25:06
%S A372270 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,
%T A372270 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,
%U A372270 0,7,9,4,9,1,6,6,8,6,8,4,6,9,4,1,1,4,2
%N A372270 Decimal expansion of the largest positive zero of the Legendre polynomial of degree 5.
%H A372270 Paolo Xausa, <a href="/A372270/b372270.txt">Table of n, a(n) for n = 0..10000</a>
%H A372270 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre_polynomials">Legendre polynomials</a>.
%H A372270 <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>.
%F A372270 Largest positive root of 63*x^4 - 70*x^2 + 15 = 0.
%F A372270 Equals sqrt(5+2*sqrt(10/7))/3.
%e A372270 0.906179845938663992797626878299392965125651910762530862873762...
%t A372270 First[RealDigits[Root[LegendreP[5, #] &, 5], 10, 100]] (* _Paolo Xausa_, Feb 27 2025 *)
%Y A372270 Cf. A008316, A100258.
%Y A372270 There are floor(k/2) positive zeros of the Legendre polynomial of degree k:
%Y A372270    k | zeros
%Y A372270   ---+--------------------------
%Y A372270    2 | A020760
%Y A372270    3 | A010513/10
%Y A372270    4 | A372267, A372268
%Y A372270    5 | A372269, A372270
%Y A372270    6 | A372271, A372272, A372273
%Y A372270    7 | A372274, A372275, A372276
%K A372270 nonn,cons
%O A372270 0,1
%A A372270 _Pontus von Brömssen_, Apr 25 2024