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
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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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
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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 *)
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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
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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