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A106284
Primes p such that the polynomial x^5-x^4-x^3-x^2-x-1 mod p has no zeros.
2
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
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
KEYWORD
nonn
AUTHOR
T. D. Noe, May 02 2005
EXTENSIONS
Name corrected by Robert Israel, Mar 13 2024
STATUS
approved