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

Author

T. D. Noe, May 02 2005

Keywords

• nonn

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

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

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)).

Programs

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]]]