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%I #18 Mar 14 2024 15:21:36
%S 1,0,0,0,0,0,0,1,2,1,0,0,0,2,3,0,2,3,1,0,1,0,0,1,1,1,1,0,1,3,1,2,3,1,
%T 1,0,0,1,1,1,1,1,1,3,1,0,1,1,1,0,0,0,0,1,0,0,2,1,0,0,3,3,1,0,1,0,0,0,
%U 1,1,1,2,1,2,0,2,0,1,1,0,1,2,0,0,2,2,1,1,2,0,0,2,1,2,2,2,1,0,0,0,0,0,0,1,0
%N Number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n).
%C This polynomial is the characteristic polynomial of the Fibonacci and Lucas 5-step sequences, A001591 and A074048. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is 9584=16*599 and 599 is the only prime for which the polynomial has 4 distinct zeros. The primes p yielding 5 distinct zeros, A106281, correspond to the periods of the sequences A001591(k) mod p and A074048(k) mod p having length less than p. The Lucas 5-step sequence mod p has one additional prime p for which the period is less than p: the 599 factor of the discriminant. For this prime, the Fibonacci 5-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^5-x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]
%o (Python)
%o from sympy import Poly, prime
%o from sympy.abc import x
%o def A106278(n): return len(Poly(x*(x*(x*(x*(x-1)-1)-1)-1)-1, x, modulus=prime(n)).ground_roots()) # _Chai Wah Wu_, Mar 14 2024
%Y Cf. A106298 (period of the Lucas 5-step sequences mod prime(n)), A106284 (prime moduli for which the polynomial is irreducible).
%Y Cf. A001591, A074048, A106281.
%K nonn
%O 1,9
%A _T. D. Noe_, May 02 2005
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