A145996 Numbers k such that quintic polynomial 4*k - k^2 + 5*k^2*x + (20*k - 20*k^2)*x^3 + (16 - 32*k + 16*k^2)*x^5 has a rational root.

0, 1, 2, 4, 243

Offset: 1

Author

Artur Jasinski, Oct 26 2008

Keywords

• nonn

Comments

When k = 1 the polynomial degenerates to degree 1.

Conjecture: This sequence is finite and complete.

This sequence is not the same as A005275 because 198815685282 does not belong to this sequence.

No more values of k less than 2*10^7.

One of the root of quintic polynomial 4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5 is Hypergeometric2F1(1/5,4/5,1/2,1/k).

Precisely for k belonging to this sequence, Hypergeometric2F1(1/5,4/5,1/2,1/k) is algebraic number of 4 degree, otherwise it is of degree 5. [Artur Jasinski, Oct 26 2008]

= sqrt(k/(k-1)) cos(3/5 arcsin(1/sqrt(k))). [Artur Jasinski, Oct 29 2008]

Crossrefs

Cf. A146160.

Programs

Mathematica

a = {}; Do[If[Length[FactorList[(4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5)]] > 2, AppendTo[a, k]; Print[k]], {k, 1, 20000000}]; a