A106281 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros.

691, 733, 3163, 4259, 4397, 5419, 6637, 6733, 8009, 8311, 9803, 11731, 14923, 17291, 20627, 20873, 22777, 25111, 26339, 27947, 29339, 29389, 29527, 29917, 34123, 34421, 34739, 34757, 36527, 36809, 38783, 40433, 40531, 41131, 42859, 43049

Offset: 1

Author

T. D. Noe, May 02 2005

Keywords

• nonn

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. The periods of the sequences A001591(k) mod p and A074048(k) mod p have length less than p.

Links

• Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Crossrefs

Cf. A106278 (number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n)), A106298, A106304 (period of Lucas and Fibonacci 5-step mod prime(n)).

Programs

Mathematica

t=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, 5000}];Prime[Flatten[Position[t, 5]]]

Python

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106281_gen(): # generator of terms
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==5:
            yield p
        p = nextprime(p)
A106281_list = list(islice(A106281_gen(),20)) # Chai Wah Wu, Mar 14 2024