OFFSET
1,5
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, May 02 2005
STATUS
approved