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%I #10 Mar 24 2024 07:55:00
%S 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,
%T 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,
%U 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
%N Number of distinct zeros of x^3-x^2-x-1 mod prime(n).
%C 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).
%H J.-P. Serre, <a href="https://doi.org/10.1090/S0273-0979-03-00992-3">On a theorem of Jordan</a>, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>.
%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, 150}]
%Y 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).
%K nonn
%O 1,5
%A _T. D. Noe_, May 02 2005
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