A106284 Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.

3, 5, 7, 11, 13, 17, 31, 37, 41, 53, 71, 79, 83, 107, 151, 157, 199, 229, 233, 239, 241, 257, 263, 277, 281, 311, 317, 331, 337, 379, 389, 409, 431, 433, 463, 467, 521, 523, 541, 547, 557, 563, 571, 577, 607, 631, 659, 677, 727, 769, 787, 809, 827, 839, 853

Offset

1,1

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

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

Maple

P:= x^5-x^4-x^3-x^2-x-1:
select(p -> [msolve(P, p)] = [], [seq(ithprime(i), i=1..10000)]); # Robert Israel, Mar 13 2024

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, 200}]; Prime[Flatten[Position[t, 0]]]

Prog

(Python)
from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106284_gen(): # generator of terms
    from sympy.abc import x
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106284_list = list(islice(A106284_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 sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).

Sequence in context: A024328 A032529 A154866 * A126145 A206864 A155801

Adjacent sequences: A106281 A106282 A106283 * A106285 A106286 A106287

Keyword

nonn

Author

T. D. Noe, May 02 2005

Extensions

Name corrected by Robert Israel, Mar 13 2024

Status

approved