A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

2, 5, 11, 13, 31, 43, 53, 79, 83, 89, 97, 103, 109, 131, 139, 151, 197, 199, 229, 233, 239, 251, 257, 271, 283, 313, 317, 347, 359, 367, 379, 389, 433, 443, 461, 479, 487, 521, 569, 571, 577, 593, 599, 601, 617, 631, 641, 643, 647, 659, 673, 677, 719, 769, 797

Offset

1,1

Comments

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.

Links

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

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

Maple

Res:= NULL: count:= 0: p:= 0:
P:= x^4 - x^3 - x^2 - x - 1:
while count < 100 do
  p:= nextprime(p);
  if [msolve(P, p)] = [] then
    Res:= Res, p; count:= count+1;
  fi
od:
Res; # Robert Israel, Mar 13 2024

Mathematica

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

Prog

(Python)
from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106283_gen(): # generator of terms
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106283_list = list(islice(A106283_gen(), 20)) # Chai Wah Wu, Mar 14 2024

Crossrefs

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p).

Sequence in context: A215214 A221868 A220141 * A020629 A355657 A224793

Adjacent sequences: A106280 A106281 A106282 * A106284 A106285 A106286

Keyword

nonn

Author

T. D. Noe, May 02 2005

Extensions

Name corrected by Robert Israel, Mar 13 2024

Status

approved