A106280 -id:A106280 - 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

A106296 Period of the Lucas 4-step sequence A073817 mod prime(n).

Original entry on oeis.org

5, 26, 312, 342, 120, 84, 4912, 6858, 12166, 280, 61568, 1368, 240, 162800, 103822, 303480, 205378, 226980, 100254, 357910, 2664, 998720, 1157520, 9320, 368872, 1030300, 10608, 1225042, 2614040, 13874, 2048382, 4530768, 136, 772880, 3307948

Offset: 1

T. D. Noe, May 02 2005

nonn

This sequence is the same as the period of Fibonacci 4-step sequence (A000078) mod prime(n) except for n=103, which corresponds to the prime 563 because the discriminant of the characteristic polynomial x^4-x^3-x^2-x-1 is -563. We have a(n) < prime(n) for primes 563 and A106280.

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

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106280 (primes p such that x^4-x^3-x^2-x-1 mod p has 4 distinct zeros), A106295.

n=4; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; 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}]

a(n) = A106295(prime(n)).

A211671 Least prime p such that the polynomial x^n - x^(n-1) - ... - 1 (mod p) has n distinct zeros.

Original entry on oeis.org

2, 11, 47, 137, 691, 25621, 59233, 2424511, 2607383, 78043403, 1032758989, 80051779

Offset: 1

T. D. Noe, Apr 18 2012

This is the characteristic polynomial of the n-step Fibonacci and Lucas sequences. For composite p, the polynomial can have more than n zeros! See A211672.

			For p = 11, x^2-x-1 = (x+3)(x+7) (mod p).
For p = 47, x^3-x^2-x-1 = (x+21)(x+30)(x+42) (mod p).
For p = 137, x^4-x^3-x^2-x-1 = (x+12)(x+79)(x+85)(x+97) (mod p).

Cf. A045468 (n=2), A106279 (n=3), A106280 (n=4), A106281 (n=5).

Cf. A211672 (for composite p).

Table[poly = x^n - Sum[x^k, {k, 0, n - 1}]; k = 1; While[p = Prime[k]; cnt = 0; Do[If[Mod[poly, p] == 0, cnt++], {x, 0, p - 1}]; cnt < n, k++]; p, {n, 5}]

a(n)={my(P=x^n-sum(k=0, n-1, x^k) ); forprime(p=2, oo, if(#polrootsmod(P,p)==n, return(p) ) );} \\ Joerg Arndt, Apr 15 2013

a(8)-a(10) from Joerg Arndt, Apr 15 2013

a(11)-a(12) from Jinyuan Wang, Apr 25 2025

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A106296 Period of the Lucas 4-step sequence A073817 mod prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A211671 Least prime p such that the polynomial x^n - x^(n-1) - ... - 1 (mod p) has n distinct zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions