Search: id:A106278
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%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, On a theorem of Jordan, Bull. Amer.
Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
%H A106278 Eric Weisstein's World of Mathematics, Fibonacci n-Step
%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
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