A371566 Primes p such that x^5 - x^4 - x^3 - x^2 - x - 1 is irreducible (mod p).

5, 7, 11, 13, 17, 31, 37, 41, 53, 79, 107, 199, 233, 239, 311, 331, 337, 389, 463, 523, 541, 547, 557, 563, 577, 677, 769, 853, 937, 971, 1009, 1021, 1033, 1049, 1061, 1201, 1237, 1291, 1307, 1361, 1427, 1453, 1543, 1657, 1699, 1723, 1747, 1753, 1759, 1787, 1801, 1811, 1861, 1877, 1997, 1999

Offset: 1

Author

Robert Israel, Mar 27 2024

Keywords

• nonn

Links

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

Crossrefs

Contained in, but not equal to, A106309. Cf. A370830.

Programs

Maple

P:= x^5 - x^4 - x^3 - x^2 - x - 1:
select(p -> Irreduc(P) mod p, [seq(ithprime(i), i=1..1000)]); # Robert Israel, Mar 13 2024

Mathematica

P = x^5 - x^4 - x^3 - x^2 - x - 1;
Select[Prime[Range[1000]], IrreduciblePolynomialQ[P, Modulus -> #]&] (* Jean-François Alcover, Mar 24 2024, after Robert Israel *)

PARI

a371566(upto) = forprime (p=2, upto, my(f=factormod(x^5 - x^4 - x^3 - x^2 - x - 1, p)); if(#f[,1]==1, print1(p,", "))) \\ Hugo Pfoertner, Mar 22 2024

Python

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A371566_gen(): # generator of terms
    p = 2
    while True:
        if Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=p).is_irreducible:
            yield p
        p = nextprime(p)
A371566_list = list(islice(A371566_gen(),20)) # Chai Wah Wu, Mar 14 2024