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

1,1

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

Table of n, a(n) for n=1..36.

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

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]]]

Prog

(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

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)).

Sequence in context: A029825 A180320 A159283 * A127341 A135316 A160500

Adjacent sequences: A106278 A106279 A106280 * A106282 A106283 A106284

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved