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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

MATHEMATICA

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]]]

PROG

(PARI)

forprime(p=2, 1000, if(#polrootsmod(x^3-x^2-x-1, p)==0, print1(p, ", ")));

/* Joerg Arndt, Jul 19 2012 */

CROSSREFS

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.

Sequence in context: A296920 A106857 A106307 * A163153 A238199 A296927

Adjacent sequences:  A106279 A106280 A106281 * A106283 A106284 A106285

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

STATUS

approved

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Last modified June 19 13:26 EDT 2019. Contains 324222 sequences. (Running on oeis4.)