A106294 -id:A106294 - cp's OEIS frontend

A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

3, 5, 23, 31, 37, 59, 67, 71, 89, 97, 113, 137, 157, 179, 181, 191, 223, 229, 251, 313, 317, 331, 353, 367, 379, 383, 389, 433, 443, 449, 463, 467, 487, 509, 521, 577, 619, 631, 641, 643, 647, 653, 661, 691, 709, 719, 727, 751, 797, 823, 829, 839, 859, 881

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644.

Primes of the form 3x^2+2xy+4y^2 with x and y in Z. - T. D. Noe, May 08 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..300
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Primes in A028952.
Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (period of Lucas and Fibonacci 3-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible mod p).
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 200}];Prime[Flatten[Position[t, 0]]]

forprime(p=2,1000,if(#polrootsmod(x^3-x^2-x-1,p)==0,print1(p,", ")));
/* Joerg Arndt, Jul 19 2012 */

A106302 Period of the Fibonacci 3-step sequence A000073 mod prime(n).

Original entry on oeis.org

4, 13, 31, 48, 110, 168, 96, 360, 553, 140, 331, 469, 560, 308, 46, 52, 3541, 1860, 1519, 5113, 5328, 3120, 287, 8011, 3169, 680, 51, 1272, 990, 12883, 5376, 5720, 18907, 3864, 7400, 2850, 8269, 162, 9296, 2494, 32221, 10981, 36673, 4656, 3234, 198, 5565

Offset: 1

T. D. Noe, May 02 2005, Sep 18 2008

nonn

This sequence differs from the corresponding Lucas sequence (A106294) at n=1 and 5 because these correspond to the primes 2 and 11, which are the prime factors of -44, the discriminant of the characteristic polynomial x^3-x^2-x-1. We have a(n) < prime(n) for the primes in A106279.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106273, A106279, A106294, A046738.

n=3; Table[p=Prime[i]; a=Join[{1},Table[0,{n-1}]]; a=Mod[a,p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i,60}]

from itertools import count
from sympy import prime
def A106302(n):
    a = b = (0,)*2+(1 % (p:= prime(n)),)
    for m in count(1):
        b = b[1:] + (sum(b) % p,)
        if a == b:
            return m # Chai Wah Wu, Feb 27 2022

a(n) = A046738(prime(n)).

A106279 Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 distinct zeros.

Original entry on oeis.org

47, 53, 103, 163, 199, 257, 269, 311, 397, 401, 419, 421, 499, 587, 599, 617, 683, 757, 773, 863, 883, 907, 911, 929, 991, 1021, 1087, 1109, 1123, 1181, 1237, 1291, 1307, 1367, 1433, 1439, 1543, 1567, 1571, 1609, 1621, 1697, 1699, 1753, 1873, 1907, 2003

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 3-step sequences, A000073 and A001644. The periods of the sequences A000073(k) mod p and A001644(k) mod p have length less than p. For a given p, let the zeros be a, b and c. Then A001644(k) mod p = (a^k+b^k+c^k) mod p. This sequence is the same as A033209 except for the initial term.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A106276 (number of distinct zeros of x^3-x^2-x-1 mod prime(n)), A106294, A106302 (periods of the Fibonacci and Lucas 3-step sequences mod prime(n)).

t=Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 500}];Prime[Flatten[Position[t, 3]]]

cp's OEIS Frontend

A106282 Primes p such that the polynomial x^3-x^2-x-1 mod p has no zeros; i.e., the polynomial is irreducible over the integers mod p.

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A106302 Period of the Fibonacci 3-step sequence A000073 mod prime(n).

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A106279 Primes p such that the polynomial x^3-x^2-x-1 mod p has 3 distinct zeros.

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Mathematica