OFFSET
1,9
COMMENTS
This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is 9584=16*599 and 599 is the only prime for which the polynomial has 4 distinct zeros. The primes p yielding 5 distinct zeros, A106281, correspond to the periods of the sequences A001591(k) mod p and A074048(k) mod p having length less than p. The Lucas 5-step sequence mod p has one additional prime p for which the period is less than p: the 599 factor of the discriminant. For this prime, the Fibonacci 5-step sequence mod p has a period of p(p-1).
LINKS
J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
MATHEMATICA
Table[p=Prime[n]; cnt=0; Do[If[Mod[x^5-x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
PROG
(Python)
from sympy import Poly, prime
from sympy.abc import x
def A106278(n): return len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=prime(n)).ground_roots()) # Chai Wah Wu, Mar 14 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, May 02 2005
STATUS
approved