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A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817.
LINKS
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
KEYWORD
nonn
AUTHOR
T. D. Noe, May 02 2005
EXTENSIONS
Name corrected by Robert Israel, Mar 13 2024
STATUS
approved

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Last modified July 24 09:00 EDT 2024. Contains 374575 sequences. (Running on oeis4.)