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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..47.

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

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, 500}]; Prime[Flatten[Position[t, 3]]]

Crossrefs

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)).

Sequence in context: A243431 A141279 A155139 * A275022 A355601 A048581

Adjacent sequences: A106276 A106277 A106278 * A106280 A106281 A106282

Keyword

nonn

Author

T. D. Noe, May 02 2005

Status

approved