The Galois group of xn−xn−1−⋯−x−1

Abstract

In this paper we prove that if n is an even integer or a prime number, then the Galois group of x n −x n−1 −· · ·−x −1 is the symmetric group Sn. This polynomial family arises quite naturally from a kind of generalized Fibonacci sequence. In order to prove our result for n = p prime, we had to prove that x p − x p−1 − · · · − x − 1 is irreducible in Fp[x], which seems to be a result of independent interest.

On m-step Fibonacci sequence in discrete Lotka-Volterra system

“…It is proved in [12] that the continued fraction (23) with b = 1 is just equal to the golden ratio τ . So, the discussion for the case where b = 1 in [1] properly coincides with that for the case where m = 2 in [6].…”

“…. , F m−1 [6]. The 2-step Fibonacci sequence is well-known as the basic Fibonacci sequence such that the ratio F j +1 /F j converges to the golden ratio τ = ( √ 5 + 1)/2 as j → ∞ [12].…”

“…The ratio of two successive and sufficiently large 3-step Fibonacci numbers approaches to the irrational number 1.839286755214161 · · · . According to [6], in an m-step Fibonacci sequence, the ratio F k+1 /F k converges to the constant c m as k → ∞, and c m is equal to one of the real solutions to the m-degree algebraic equation…”

“…However, in [4,5,11], the function a (n) j is not mainly discussed. In this paper, by focusing on a (n) j , we first associate the determinant solution (3) with the m-step Fibonacci sequences [6] which cover the well-known Fibonacci, Tribonacci sequences [12] and so on. This study is mainly twofold.…”

Diophantine Triples and k-Generalized Fibonacci Sequences

“…Proof. The cases k ≡ 1, 2 (mod 4) have already been treated both in [2] and in [16]. We treat the remaining cases.…”