A106276 Number of distinct zeros of x^3-x^2-x-1 mod prime(n).

1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 3, 3, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 3, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 3, 3, 1, 3, 3, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1

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 recursions, A000073 and A001644. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -44 = -4*11. The primes p yielding 3 distinct zeros, A106279, correspond to the periods of the sequences A000073(k) mod p and A001644(k) mod p having length less than p. The Lucas 3-step sequence mod p has two additional primes p for which the period is less than p: 2 and 11, which are factors of the discriminant -44. For p=11, the Fibonacci 3-step sequence mod p has a period of p(p-1).

Links

• 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.

Crossrefs

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106293 (period of the Lucas 3-step sequences mod prime(n)), A106282 (prime moduli for which the polynomial is irreducible).

Programs

Mathematica

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]