A106277 Number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n).

0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 2, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 2, 2, 2, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 2, 1, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 0, 0

Offset: 1

Author

T. D. Noe, May 02 2005

Keywords

• nonn

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -563 and 563 is the only prime for which the polynomial has 3 distinct zeros. The primes p yielding 4 distinct zeros, A106280, correspond to the periods of the sequences A000078(k) mod p and A073817(k) mod p having length less than p. The Lucas 4-step sequence mod p has one additional prime p for which the period is less than p: the discriminant 563. For this prime, the Fibonacci 4-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

Crossrefs

Cf. A106296 (period of the Lucas 4-step sequences mod prime(n)), A106283 (prime moduli for which the polynomial is irreducible).

Programs

Mathematica

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]

Python

from sympy.abc import x
from sympy import Poly, prime
def A106277(n): return len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=prime(n)).ground_roots()) # Chai Wah Wu, Mar 29 2024