Solvable sextics and Fibonacci numbers

While doing some research on solvable sextics, I stumbled upon this
unusual connection between it and Fibonacci/Lucas numbers.

Define the sequence, starting with n=0:

L_n = {2, 3, 7, 18, 47, 123, 322,...}

This is A005248 in the OEIS (Online Encyclopedia of Integer Sequences)
and has formula:

L_n = phi^(2n) + (1/phi)^(2n)

where phi is the golden ratio = (1+Sqrt[5])/2.

Conjecture: "The irreducible sextic x^6 - ( L_n)x^5 + (L(n+1))x - 1 =
0 is solvable in radicals and factors over the extension Sqrt[5]."

Thus, x^6-123x^5+322x-1 = 0 is solvable, and so on.


Anybody knows how to prove the conjecture?

Tito
.

Relevant Pages

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