A106283 -id:A106283 - cp's OEIS frontend

A106277 Number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n).

0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 2, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 2, 2, 2, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 2, 1, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 0, 0

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -563 and 563 is the only prime for which the polynomial has 3 distinct zeros. The primes p yielding 4 distinct zeros, A106280, correspond to the periods of the sequences A000078(k) mod p and A073817(k) mod p having length less than p. The Lucas 4-step sequence mod p has one additional prime p for which the period is less than p: the discriminant 563. For this prime, the Fibonacci 4-step sequence mod p has a period of p(p-1).

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106296 (period of the Lucas 4-step sequences mod prime(n)), A106283 (prime moduli for which the polynomial is irreducible).

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, 150}]

from sympy.abc import x
from sympy import Poly, prime
def A106277(n): return len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=prime(n)).ground_roots()) # Chai Wah Wu, Mar 29 2024

A370830 Primes p such that the polynomial x^4-x^3-x^2-x-1 is irreducible mod p.

Original entry on oeis.org

2, 5, 31, 43, 53, 79, 83, 89, 97, 109, 131, 139, 151, 199, 229, 233, 239, 283, 313, 317, 359, 367, 389, 433, 443, 479, 487, 569, 571, 577, 601, 617, 641, 643, 659, 677, 769, 797, 823, 853, 857, 929, 937, 941, 971, 1013, 1019, 1049, 1063, 1069, 1087, 1093, 1117, 1163, 1171, 1181, 1231, 1249, 1283

Offset: 1

Robert Israel, Mar 13 2024

nonn

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

Subsequence of A106283. Cf. 106309.

P:= x^4 - x^3 - x^2 - x - 1:
select(p -> Irreduc(P) mod p, [seq(ithprime(i),i=1..1000)]);

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A370830_gen(): # generator of terms
    p = 2
    while True:
        if Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).is_irreducible:
            yield p
        p = nextprime(p)
A370830_list = list(islice(A370830_gen(),20)) # Chai Wah Wu, Mar 14 2024

cp's OEIS Frontend

A106277 Number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n).

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