cp's OEIS Frontend

A106278 Number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n).

%I A106278 #19 Feb 16 2025 08:32:57
%S A106278 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 A106278 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 A106278 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 A106278 Number of distinct zeros of x^5-x^4-x^3-x^2-x-1 mod prime(n).
%C A106278 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 A106278 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 A106278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Fibonaccin-StepNumber.html">Fibonacci n-Step Number</a>
%t A106278 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 A106278 (Python)
%o A106278 from sympy import Poly, prime
%o A106278 from sympy.abc import x
%o A106278 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 A106278 Cf. A106298 (period of the Lucas 5-step sequences mod prime(n)), A106284 (prime moduli for which the polynomial is irreducible).
%Y A106278 Cf. A001591, A074048, A106281.
%K A106278 nonn
%O A106278 1,9
%A A106278 _T. D. Noe_, May 02 2005