%I A106278
%S A106278 1,0,0,0,0,0,0,1,2,1,0,0,0,2,3,0,2,3,1,0,1,0,0,1,1,1,1,0,1,3,1,2,3,1,1,
%T A106278 0,0,1,1,1,1,1,1,3,1,0,1,1,1,0,0,0,0,1,0,0,2,1,0,0,3,3,1,0,1,0,0,0,1,1,
%U A106278 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="http://www.ams.org/bull/2003-40-04/S0273-0979-03-00992-3/
               S0273-0979-03-00992-3.pdf">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="http://mathworld.wolfram.com/
               Fibonaccin-Step.html">Fibonacci n-Step</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}]
%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 Sequence in context: A004198 A116402 A093323 this_sequence A064287 A128206 
               A103294
%Y A106278 Adjacent sequences: A106275 A106276 A106277 this_sequence A106279 A106280 
               A106281
%K A106278 nonn
%O A106278 1,9
%A A106278 T. D. Noe (noe(AT)sspectra.com), May 02 2005