A106278 Number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n).

1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 2, 3, 0, 2, 3, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 1, 2, 3, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 3, 1, 0, 1, 0, 0, 0, 1, 1, 1, 2, 1, 2, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 2, 2, 1, 1, 2, 0, 0, 2, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0

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

Table of n, a(n) for n=1..105.

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

Cf. A106298 (period of the Lucas 5-step sequences mod prime(n)), A106284 (prime moduli for which the polynomial is irreducible).

Cf. A001591, A074048, A106281.

Sequence in context: A324351 A116402 A093323 * A339829 A177517 A227819

Adjacent sequences: A106275 A106276 A106277 * A106279 A106280 A106281

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved